Irving  Stringham 


WENTWORTH'S 
SERIES    OF     MATHEMATICS. 


First  Steps  in  Number. 

Primary  Arithmetic. 

Grammar  School  Arithmetic. 

High  School  Arithmetic. 

Exercises  in  Arithmetic. 

Shorter  Course  in  Algebra. 

Elements  of  Algebra.  Complete  Algebra. 

College  Algebra.  Exercises  in  Algebra. 

Plane  Geometry. 

Plane  and  Solid  Geometry. 

Exercises  in  Geometry. 

PI.  and  Sol.  Geometry  and   PI.  Trigonometry. 

Plane  Trigonometry  and  Tables. 

Plane  and  Spherical  Trigonometry. 

Surveying. 

PI.  and  Sph.  Trigonometry,  Surveying,  and  Tables. 

Trigonometry,  Surveying,  and  Navigation. 

Trigonometry  Formulas. 

Logarithmic  and  Trigonometric  Tables  (Seven). 

Log.  and  Trig.  Tables  {Complete  Edition), 

Analytic  Geometry. 


Special  Terms  and  Circular  on  Application, 


ALGEBIUIC  ANALYSIS. 

■  solutions  and   exercises 


ILLUSTRATING 


THE  FUNDAMENTAL  THEOREMS  AND  THE 

MOST  IMPORTANT  PROCESSES  OF 

PURE  ALGEBRA. 


BY 

G.  A.  WENTWORTH,  A.M., 

Professor  or  Mathematics  in  Phillips  Exeter  Academy: 

J.  A.  McLELLAN,  LL.D., 

Inspector  or  Normal  Schools,  and  Conductor  op 
Teachers'  Institutes,  for  Ontario,  Canada; 


J.  c.  glashan, 

Inspector  of  Public  Schools,  Ottawa,  Canada, 


PART  I. 


BOSTON,  U.S.A.: 

PUBLISHED   BY   GINN   &   COMPANY, 

1889. 


9f^3 


/" 


Entered,  according  to  Act  of  Congress,  in  the  year  1889,  by 

G.  A.  WENTWORTH, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


All  Rights  Reserved. 


Typography  by  J.  S.  Cushing  &  Co.,  Boston,  U.S.A. 


Presswork  by  Ginn  &  Co.,  Boston,  U.S.A. 


PREFACE. 


rriHE  work  of  which  this  volume  forms  the  first  or  introductory 
part  is  intended  to  supply  students  of  mathematics  with  a  well- 
filled  storehouse  of  solved  examples  and  unsolved  exercises  in  the 
application  of  the  fundamental  theorems  and  processes  of  pure  Alge- 
bra, and  to  exhibit  to  them  the  highest  and  most  important  results 
of  modern  algebraic  analysis.  It  may  be  used  to  follow  and  sup- 
plement the  ordinary  text-books,  or  it  may  be  employed  as  a 
guide-book  and  work  of  reference,  in  a  course  of  instruction  under 
a  teacher  of  mathematics. 

The  following  are  some  of  the  special  features  of  this  volume  : 

It  gives  a  large  number  of  solutions  in  illustration  of  the  best 
methods  of  algebraic  resolution  and  reduction,  some  of  which  are  not 
found  in  any  text-book. 

It  gives,  classified  under  proper  heads  and  preceded  by  type- 
solutions,  a  great  number  of  exercises,  many  of  them  illustrating 
methods  and  principles  which  are  generally  ignored  in  elementary 
Algebras  ;  and  it  presents  these  solutions  and  exercises  in  such  a 
way  that  the  student  not  only  sees  how  algebraic  transformations 
are  effected,  but  also  perceives  how  to  form  for  himself  as  many 
additional  examples  as  he  may  desire. 

It  shows  the  student  how  simple  principles  with  which  he  is  quite 
familiar,  may  be  applied  to  the  solution  of  questions  which  he  has 
thought  beyond  the  reach  of  these  principles ;  and  gives  complete 
explanations  and  illustrations  of  important  topics  which  are  omitted 
or  are  barely  touched  upon  in  the  ordinary  books,  such  as  the  Prin- 


800573 


IV  PREFACE. 


ciple  of  Symmetry,  Theory  of  Divisors,  and  its  application  to  Fac- 
toring, and  Applications  of  Horner's  Division. 

A  few  of  the  exercises  are  chiefly  supplementary  to  those  proposed 
in  the  text-books,  but  the  intelligent  student  will  find  that  even 
these  examples  have  not  been  selected  in  an  aimless  fashion  ;  he  will 
recognize  that  they  are  really  expressions  of  certain  laws  ;  they  are 
in  fact  proposed  with  a  view  to  lead  him  to  investigate  these  laws 
for  himself  as  soon  as  he  has  sufficiently  advanced  in  his  course. 
Nos.  8,  9,  10,  and  11  of  Ex.  1  afford  instances  of  such  exercises. 

Others  of  the  questions  proposed  are  preparatory  or  interpretation 
exercises.  These  might  well  have  been  omitted  were  it  not  that  they 
are  generally  omitted  from  the  text-books  and  are  too  often  neglected 
by  teachers.  Practice  in  the  interpretation  of  a  new  notation,  and 
in  expression  by  means  of  it,  should  always  precede  its  use  as  a  sym- 
bolism itself  subject  to  operations.  Nos.  23  to  36  of  Ex.  3,  and  nearly 
the  whole  of  Ex.  15,  may  serve  for  instances. 

By  far  the  greater  number  of  the  exercises  is  intended  for  prac- 
tice in  the  methods  exhibited  in  the  solved  examples.  As  many  as 
possible  of  these  have  been  selected  for  their  intrinsic  value.  They 
have  been  gathered  from  the  works  of  the  great  masters  of  analysis, 
and  the  student  who  proceeds  to  the  higher  branches  of  mathematics 
will  meet  again  with  these  examples  and  exercises,  and  will  find 
his  progress  aided  by  his  familiarity  with  them,  and  will  not  have 
to  interrupt  his  advanced  studies  to  learn  theorems  and  processes 
properly  belonging  to  elementary  Algebra.  In  making  this  selec- 
tion, it  has  been  found  that  the  most  widely  useful  transformations 
are,  at  the  same  time,  those  that  best  exhibit  the  methods  of  reduc- 
tion here  explained,  so  that  they  have  thus  a  double  advantage. 

The  present  volume  ends  with  an  extensive  collection  of  exercises 
in  Determinants.  These  present  under  new  forms  and  from  a  dif- 
ferent point  of  view  the  greater  number  of  the  theorems  proposed, 
and  many  of  the  general  results  obtained,  in  the  earlier  chapters,  and 
to  these  they  add  many  important  propositions  in  other  subjects ;  as, 


PREFACE. 


for  example,  in  the  method  of  least  squares,  in  linear,  homograph ic, 
orthogonal,  and  homaloid  transformations,  and  in  the  degeneracy 
and  the  tangency  of  quadrics. 

The  second  volume  will  treat  of  factorials  and  the  combinatory 
analysis ;  finite  differences  and  derived  functions,  both  direct  and 
inverse,  of  explicit  functions  of  a  single  variable  ;  expansion,  sum- 
mation, reversion,  transformation,  and  interpolation  of  series  ;  the 
arithmetic,  harmonic,  and  geometric  series  of  integral  orders,  includ- 
ing the  theta-functions  ;  recurring  series  ;  binomial,  logarithmic,  and 
exponential  series  ;  hyperbolic  and  circular  functions  ;  trigonometric 
series,  direct  and  inverse  ;  Legendre's,  BessePs,  Lame's,  and  Heine's 
series  and  their  associated  functions  ;  double  series  ;  infinite  prod- 
ucts ;  continued  fractions  ;  indeterminate  equations ;  theory  of  num- 
bers ;  inequalities  ;  maxima  and  minima ;  binomial  equations  and 
cyclotomic  functions  ;  transformation  of  binary  forms  ;  theory  of  the 
quintic  and  of  higher  equations  ;  theory  of  substitutions.  The  whole 
will  close  with  a  chapter  on  the  fundamental  postulates  and  the 
general  laws  of  algebra,  illustrated  by  examples  and  problems  in 
matrices,  polar  algebras,  and  ideal  arithmetic. 

In  this  second  part  of  the  work  the  authors  hope  to  be  able  to 
give  numerous  historical  notes  and  bibliographical  references  for 
the  use  of  students  who  desire  to  pursue  the  subject  further,  or 
to  consult  the  original  memoirs. 

A  companion  volume  to  the  present  is  in  course  of  preparation  for 
the  use  of  private  students  and  of  all  who  have  not  the  advantage 
of  instruction  by  a  specialist  in  mathematics.  The  companion  will 
contain  proofs  of  the  theorems  employed  and  solutions  of  the  exer- 
cises proposed  in  this  volume,  the  whole  accompanied  by  hints  on 
the  best  method  of  attacking  problems,  and  on  the  selection  of  pro- 
cesses for  their  reduction. 

Notwithstanding  that  the  utmost  care  has  been  taken  in  revis- 
ing the  proof-sheets,  there  doubtless  remain  many  errors  both  in 
the  examples  and  in  the  exercises.     The  authors  would  feel  grate- 


VI  PREFACE. 


ful  to  teachers  and  students  for  notification  of  all  errors  which 
may  be  discovered,  and  also  for  suggestions  in  relation  to  the  im- 
provement of  the  work. 

Messrs.  J.  S.  Gushing  &  Co.  deserve  special  mention  for  their 
masterly  skill  in  overcoming  all  the  difficulties  in  the  typography 
of  this  work,  and  for  their  excellent  taste  and  judgment  exhibited 
in  the  beauty  and  elegance  of  these  pages. 

G.    A.   WENTWORTH. 

J.   A.    McLELLAN. 

J.    C.    GLASHAN. 


Note.    It  is  due  Mr.  Glashan  to  state  that  the  main  part  of  the 
work  on  this  Algebra  has  been  done  by  him. 

G.   A.  Wentworth. 
J.   A.    McLellan. 


CONTENTS. 


CHAPTER   I. 

Substitution,  Horner's  Division,  Etc.  p^^^^ 

Numerical  and  Literal  Substitution 1 

Fundamental  Formulas  and  their  Applications  ...       13 

Expansion  of  Binomials 26 

Horner's  Methods  of  Multiplication  and  Division,  and  their 
Applications 27 

CHAPTER   II. 

Principle  of  Symmetry,  Etc. 

The  Principle  of  Symmetry  and  its  Applications         .         .      39 
The  Theory  of  Divisors  and  its  Applications       ...      49 


Factorimo.  chapter    III. 

Direct  Application  of  the  Fundamental  Formulas       .         .  77 

Extended  Application  of  the  Formulas        ....  88 

Factoring  by  Parts 100 

Application  of  the  Theory  of  Divisors          ....  105 

Factoring  a  Polynome  by  Trial  Divisors     ....  114 


CHAPTER   IV. 

Measures  and  Multiples,  Etc. 

Division,  Measures,  and  Multiples 130 

Fractions 140 

Ratios 155 

Complete  Squares,  Cubes,  Etc 164 


Vlll  CONTENTS. 


CHAPTER   V. 
Linear  Equations  of  One  Unknown  Quantity. 

PAGE 

Preliminary  Equations 171 

Fractional  Equations 173 

Application  of  Ratios 179 

Resolution  by  Rejection  of  Constant  Factors       .         .         .  183 
Higher  Equations  which   are    Resolvable   into    Rational 

Linear  Factors 191 


CHAPTER   VL 
Simultaneous  Linear  Equations. 

Systems  of  Equations 202 

Application  of  Symmetry 207 

Particular  Systems  of  Linear  Equations      .         .         .         .211 

CHAPTER   VII. 
Quadratic  Equations. 

Pure  Quadratics 226 

Quadratic  Equations  and  Equations  that  can  be  solved  as 

Quadratics 231 

Simultaneous  Quadratic  Equations 238 


CHAPTER   VIII. 
Indices  and  Surds. 

Indices  and  Surds 263 

Complex  Quantities 273 

Surd  Equations 277 


CHAPTER  IX. 
Cubic  and  Quartic  Equations 

Cubic  Equations  ........     297 

Quartic  Equations 305 


CONTENTS. 


IX 


CHAPTER  X. 


Determinants. 


Definitions  and  Notation     . 
Transformation  of  Determinants 
Evolution  of  Determinants . 
Multiplication  of  Determinants  . 
Applications  of  Determinants 


PAQE 

315 
321 
332 
342 
348 


ALGEBEA. 


CHAPTER   I.  —  SUBSTITUTION. 

Ex.  1. 

1.    If  a=l,   h  =  2,  c=-3,  d  =  ^,  x  =  S),  y  =  8,    find  the 
values  of  the  following  expressions  : 


i-fi-(i-i-a-)r, 

a  —  {x  —  y)  —  (h  —  c)(d~  a)  ~{y  —  h)  {x  +  c) ; 

{x  +  d){y  +  h  +  c)  +  {x~d){a--h  —  d) 

-\-  {y  +  d){a  —  X  —  d)] 
{d~xy  +  (c  +  yy; 
(a  -  b)  (c'  ~b'x)-(c-  d)  (b'  -  a'x) 

+  (d-b-c)(d'-a'); 
d  —  a.d-\~c_c^d-{-b 
c?  +  a      d  —  c         d —  b 

2.    If  a  =  3,  b  =  —4:,  c=—9,  and  2s-=^a+b  i- c,   find 
the  values  of  the  following  expressions  : 

s(s  —  d)(s  —  b)  (s  ~  c)] 
s'  +  (s  -  af  +  (s  -  by  +(s~  cf  ; 
s^  —  (s  —  a)  (s  —  b)  —  (s  —  b)(s  —  c)  —  (s  —  c)  (s  —  a) ; 
2(5  —  a)  (5  —  b)  (s  —  c)  +  a(s  —  b)  (s  -  c) 
-\-^b(s  —  c)  (5  —  a)  -f  c  (s  —  a)  (s  —  by. 


SUBSTITUTION. 


3.    If  a  —  12,  3  :^  —  3,  c  =  1,  X  =  4^,  find  the  values  of  the 
following  expressions : 

a'-h',     a'  +  b\     (a~b)\     (a  -  hf . 
a'  +  b''     a'-b''     {a  +  bf     {a  +  bf' 

a^  +  ab  +  b\     a^  —  b\     x  j  2x-J  __^x--l  \  x ~-l 

3  4 


21 


(a  +  b)l(a  +  by-c'\  .     dXb-c)+bXc~a)+c\a-b) 
W'c'-(a:'~b'-cJ  '  (a-  6)  (^  -  e)  (c-a) 


4.  If  a  —  6,  ^  ==  5,  c  =  —  4:,  d  =  —  3,  find  the  values  of  the 

following  expressions : 

V(^^  +  ac)  +  V(^-' -2ac)',    -^\b'  +  ac+  ^(c'-2 ac) \ ; 
2 a  -  V(^'  -  «^)  '     c^  +  2d{d''~  c') 

5.  If  :r  =  3,  ?/  =  4,  2;  —  0,  find  the  values  of 

\zx  -  v(^^  +  fWfi^  +  V(*'  +  f+^'n  ; 

a,-"  +  f+x';     {x  -  yj-"  +  (y  -  z)'""  +  («  -  x)'"" ; 


6.    Fmd  the  values  of     (^  +  ^  +  .)-- 3(.-  + y-+ z^) 
when 
(i.)    x=\,     y  =  %     2-3 
(ii.)    a; -2,      y-3,      2-4 
(iii.)    a; -3,      y-4,      2-5 
(iv.)    :t'-10,   y-11,    2-12. 


SUBSTITUTION. 


7.  Given  rr  =  3,  y  =  4,  z  =  —  5,  find  the  values  of 

{x  +  y^+  xj—  3  {x  +  y  +  z)  {xij  +  y^  +  zx)  ; 

x'iy  +  z)  +  y'{z  +  r^O  +  2'(:^  +  y)  +  2:z:yz  ; 

x'(7j  ~-z)  +  y\z  -  ^)  +  2^^  ~  y) ; 

ihx  -  4 2)^  +  9(4:r  -  zj  -  (13^  -  5^7  ; 

(3a; +  4y  +  5^)^+(4^  +  3y  +  12^)*^-(5a;  +  5y+ 13^)1 

8.  If  s  =  a  +  ^  +  <^,  find  the  values  of 

(2s  -  af-\-  (2s  -  hj  -  (2s  +  cf 
when 

(i.)  a  =  3,  5  ==4,  ^  =  5; 

(ii.)  a -=21,  ^  =  20,  ^  =  29; 

(iii.)  a -119,  ^=-120,  c-:169; 

(iv.)  a  =  3,  h^-\,  c--=5; 

(v.)  a  =--5,  Z>---12,  6' --13. 

9.  If  a=-l,  h  =  S,  c=5,  d=7,  e  =  9,f==-U,  show  that 


f/=(«i 
I+l  +  l+i  +  i^lA^iV 

lb      he      ccl     de      ef      2\a     /y 


ah 

a^c      hcd      cde      def      4\a^      ^7  ' 

ahcd     hcde      cdef      6\abc      defy 

a'  +  h'  +  c'  -  ah -he  —  ca  -  Z>'+  c''+  c^^—  Z;^ — cd~-  dh 
—  c''^  -\'  d^^  -\-  e^  ~  cd  ~  dc  —  ^c 


SUBSTITUTION. 


10.    If  a  =  l,   b  =  2,   c  =  3,    d  =  4:,    e  =  5,  /- 6,   g=7, 
show  that 

a-\-  b  -{-  c=  icd]     a-\- b -}- c -{- d=  ide; 

a  +  b  +  c  +  d  +  e=ief;    a  +  b  +  c  +  d+ e+f=--ifg; 

a^  _^_  b-'-Yc^^^  ^^('^  +  '^)  ; 
ab  (a  +  b) 

«3  +  j.  +  ,.+  ^.=.^feM+|}. 

ab  (a  +  b) 

a^  +  b'^  +  c^  +  d'  +  >?=  'ffy,?; 

ab{a-{-  b) 

^2      I      7,2      I         2  _l      ^2      I        2      I      p  _   /^  (y    +  ^)   . 

a^j^b'  +  c'^{a  +  b  +  cy; 

oJ^  +  h'  +  c'  +  d^  =  (a  +  b  +  c  +  dy  ', 

d  +  b'  +  c'  +  d'  +  e'  =  (a  +  b  +  c  +  d+  ef  ; 

^3  _!_  53  _^  ^3  _!_  ^3  _^  ^3  j^p  ^.(^a-^i_Yc^d-\-e-\-fY', 

^.j^l.j^,^^^cd{c-\-d){^d-V), 
bc{b  +  e) 

«* + 6*  +  c' +  ^^  =  *i^±#^%^^ 

6(?  (0  +  c) 

„.  +  ..  +  ..  +  ..  +  ,+y.^.W+g(;&-i); 


SUBSTITUTION. 


11.    Assume  any  numerical  values  for  x,  y,  and  z,  and  find 
the  values  of  the  following  expressions  : 

(a;^  -  10a:'  +  ^xf  +  (5«*  -  IO3?  +  1)'  -  {a?  +  Vf ; 

{x  +Yf-2{x+  5)'  -  (x  +  9/  +  2(x  +  11/ 
+  (:r+12/-(a.-+16)^ 

{x'~yj  +  {2xyy-{x'^  +  yj; 

(x'  -  SxyJ  +  (Sx'y  ~  yj  -  {x'  +  ff  ■ 

(3  «^  +  4  xy  +  yj  +  (4  a;'  +  2  xy^  -  (5  a;'  +  4  a-y + y^)^ ; 

(x-yy  +  {y-zf+{z-xy-2.{x~y){y-z){z-x). 

§  1.  If  :r  —  any  number  (as,  for  example,  3),  then 
x\==  xXx)-=  Zx\  0(?{=  xXx'')  =  Zx^\  x\=  X  X  x^)  =  2>x\ 
etc.  Or,  3  =  0;;  Zx^x^\  ?>x^  —  x^\  3^*  =  :r^;etc.  Hence, 
problems  like  the  following  may  be  solved  like  ordinary 
arithmetical  problems  in  "Keduction  Descending." 

Examples. 
1.    Find  the  value  of  x^  —  2x--^,  when  x  =  5. 

5 


5x 

—  2x 

Sx 
5 


I 


15 

-9 

6  Ans. 


6  SUBSTITUTION. 

2.  Find  the  value  of  x^  —  x^  —  4:x'^  —  ^^ x  —  o  when  x  =  o. 

x'-x'  —  4:x'~Sx-5 
3 

Pi 3  ^^ 

-~x^ 

n ^x^ 

3 

P2 ^X^ 

~4:X^ 

r, 2x'' 

3 

—  3x 

rg 3^ 

3 

p. 9 

-5 

r^ 4  Ans. 

3.  Find  the  value  of  2a;^+12^'+6:i;'— 12:^;+10  when  ^  =  -5. 

Using  coefficients  only,  we  have 

2  +  12  +  6-12  +  10 
-5 

Pi -10 

+  12 

n +  2 

-  5 

P2 -10 

+   6 
^2 —  4 

-  5 

i93 20 

-  12 

n 8 

-  5 

p. -40 

+  10 

7*4 —  30  Ans. 


\ 


SUBSTITUTION. 


§  2.  If  the  coefficients,  and  also  the  values  of  x,  are  small 
numbers,  much  of  the  above  may  be  done  mentally,  and  the 
work  will  then  be  very  compact.  Thus,  performing  men- 
tally the  multiplications  and  additions  (or  subtractions)  of 
the  coefficients,  and  merely  recording  the  partial  reductions 
^'i,  ^2,  ^^3)  and  the  result  ^4,  the  last  example  will  appear  as 


follows : 


-5)2 

2 

-4 

8 

-30 


+  12    +6    -12    +10 


§  3.  In  the  above  examples,  the  coefficients  are  "brought 
down  "  and  written  below  the  products  ^1,  ^2,  Ps,  Pi,  and  are 
added  or  subtracted,  as  the  case  may  require,  to  get  the 
partial  reductions  Ti,  r^,  rg,  and  the  result  r^.  Instead  of 
thus  "bringing  down"  the  coefficients,  we  may  "carry  up" 
the  products  jp^,  p^,  p-^,  />4,  writing  them  beneath  their  cor- 
responding coefficients,  and  thus  get  ri,  i\,  i\,  r^,  in  a  third 
(horizontal)  line.  Arranged  in  this  way,  Exam.  2  will 
aj)pear 


1    -1    -4 

+  3    +6 

-3 

+  6 

-5 
+  9 

1     +2    +2 

+  3; 

4 

and  Exam.  3  will  appear 


-5 


2+12 

-10 

+  6 
-10 

-12    +10 
+  20    -40 

2      +2 

-4 

+  8;   -30 

Comparing  these  arrangements  with  those  first  given 
(Exams.  2  and  3),  it  will  be  seen  that  they  are,  figure  for 
figure,  the  same,  except  that  the  multiplier  is  not  repeated. 


SUBSTITUTION. 


§  4.  When  there  are  several  figures  in  the  value  of  x, 
they  may  be  arranged  in  a  column,  and  each  figure  used 
separately,  as  in  common  multiplication.  When  only  ap- 
proximate values  are  required,  "  contracted  multiplication  " 
may  be  used. 

4.    Find  the  value  of  3.x'^-160^^  +  344^'^  +  700^'-1910:r 


+ 1200,  given  x  =  51. 


1 

50 

3 

-160 

3 

150 

+  344 

-7 

-350 

+  700 

-13 

-650 

-1910 

37 
1850 

+  1200 

-23 

-1150 

3 

-7 

-13 

+  37 

-23; 

+  27 

64   0 

0 

-144 

+  45 

64 

75.712 

89.5673 

-  38.0419 

6.4 

7.5712 

8.9567 

-  3.8042 

5.12 

6.0570 

7.1654 

-  3.0434 

0.192 

0.2271 

0.2687 

-0.1141 

64  75.712 

89.5673 

-  38.0419 

-  0.0036 

.*.  result  is  27. 

5.    Given  a;  =  1.183,  find  the  value  of  64:t*  -  144^' +  45 
correct  to  three  decimal  places. 


.-.result  is— 0.004. 

Ex.  2. 

Find  the  value  of 

1.  x'  -  llo;'  -  11^;*'  -  13a;  +  11,  for  x  =  12. 

2.  ^■*  + 50a;' -16a;' -16a; -61,  for  a;  =  -17. 

3.  2a;' +  249a;' -125a;' +  100,  for  a;-: -125. 

4.  2a;' -473a,-' -234a,' -711,  for  a;  =  200. 

5.  x^  —  3a;'  — 8,  for  a;:==4. 


SUBSTITUTION. 


6.  x^  -  515:r^  -  3127 :r*  +  b2bx^  -  2090:^^  +  3156a; 

—  15792,  for  a;-- 521. 

7.  2x'  +  401^*  -  199 :r"'  +  399^^  -  m2x  +  211, 

forr?;  =  — 201. 

8.  1000:?;*- 81  r^,  hi  x  =  01. 

9.  mx^  +  nix^  -  2b1x''  -  325.^  -  50,  for  x  =  If. 

10.  5x'  +  497a;'  +  200:r'  +  196r^^  -  218:?;-  2000, 

for  rr  =-  —  99. 

11.  5x'  -  620:?;*  -  1030:?;^  +  1045^;^  -  4120:?;  +  9000, 

for  X  =  205. 

Calculate,  correct  to  three  places  of  decimals : 

12.  .'?;'  + 3:?;' -13:?; -38,  for  :?;=:3.58443,  for  :?;  =  - 3.77931, 

and  for  :?;=  — 2.80512. 

13.  y*-  14y2+  y  +  38,  for  y  =  3.13131,  for  ?/=- 1.84813, 

and  for  y=:  — 3.28319. 


Ex.  3. 

What  do  the  following  expressions  become  (i.)  when  x^a; 
(ii.)  when  x= — a? 

1.  x^  —  4.ax^  +  6a''x''  —  4:a^x-}-a\ 

2.  ^(x'-ax+a').  3.    ^{x^  +  2ax  +  a'). 
4.    (x'  +  ax  +  ay  —  (x"  —  ax  +  a'f. 

If  x  =  'i/  =  z  =  a,  find  the  value  of  the  following  expres- 
sions : 


5-    i^~y){y  ~z){z~x). 

6.    {x  +  yy{y  +  z~-a){x  +  z~a). 


10  SUBSTITUTION. 


^{y  +  ^)  {f  +  ^'  —  ^') +yi^  +  ^)  (^'  +  ^'  -  f) 


X 


y+z      x+z      x+y 

Find  the  value  of 

^     X  .  X      1  ahc 

9.    -  +  7,  when  07  = '• 

a     b  a-\-  0 

10         1.1.1 


— -  +  — -+— whenr?:=-(a-^>  +  c). 

a{b  —  x)      b{c  —  x)      a(x--  c)  a 

a      o  —  a  b(b-\-a) 

12.  (a  +  ^')  {h  +  x)  —  a{h  -{-  c)-\-  x^,  when  r?;  — — -• 

13.  bx  -\-  cy  -\-  az,  when  r^  =  6  +  ^  —  ^?  y  =  c  +  a  —  b^ 

z^=a-\-b  —  c. 

^^     a{l  +  b)  +  bx a      _^hena;  =  -a. 

a{l-\-b)  —  bx     a  —  ^bx 

''•    f^T-"  +  ^"t?.when^  =  i(5-«). 
yx-^-bJ      x  —  a  —  2b 

16.  (^  — $')(^  +  2r)  +  (r  — rr)(^  +  5'),  when  a7=    ^  ^ — ^• 

17.  aX5-d?)  +  Z>2(^__^)_|_^2(^_^>j^  whena-^)  =  0. 

18.  {a  +  b  +  c) (be  +  ca  +  ab)  —  {a  +  b){b  +  c) {c  +  a), 

when  a  =  —  b. 

19.  (a  +  Z>  +  c)'—  (a'  +  b^  +  c'),  when  a  +  &  =  0. 

20.  (x  +  y  +  zy-(x  +  yy-(y  +  zy~-{z  +  xy  +  x'  +  y'  +  z\ 

when  rz;  -f  2/  +  2J  =  0- 

21.  a^(c  -  b')  +  b\a-  c')  +  c'(b  -  a')  +  abc{abc  -  1), 

when  Z>  —  a^  =  0. 


/7 


SUBSTITUTION.  11 


22.  «=  f«\+5^7+  l^  r^^Y-  when  «^  +  J^  =  0. 

23.  Express  in  words  the  fact  that  (a  —  by  =  a^~2ab-\-h'^. 

24.  Express  algebraically  the  fact  that  "the  sum  of  two 

numbers  multiplied  by  their  difference  is  equal  to 
the  difference  of  the  squares  of  the  numbers." 

25.  The  area  of  the  walls  of  a  room  is  equal  to  the  height 

multiplied  by  twice  the  sum  of  the  length  and 
breadth.  What  are  the  areas  of  the  walls  in  the 
following  cases :  (i.)  length  Z,  height  h,  breadth  b ; 
(ii.)  height  x,  length  b  feet  more  than  the  height, 
and  breadth  b  feet  less  than  the  height. 

26.  Express  in  w^ords  the  statement  that 

(x  -{-  a)  (x -}- b)  =  a^  -{- (a -\- b)  X -}-  ab. 

27.  Express  in  symbols  the  statement  that  ''the  square  of 

the  sum  of  two  numbers  exceeds  the  sum  of  their 
squares  by  twice  their  product." 

28.  Express  in  words  the  algebraic  statement, 

(x  +  yY  =  x^  +  y^  +  Sx7/(x-{-  y), 

29.  Express  algebraically  the  fact  that  "the  cube  of  the 

difference  of  two  numbers  is  equal  to  the  difference 
of  the  cubes  of  the  numbers  diminished  by  three 
times  the  product  of  the  numbers  multiplied  by 
their  difference." 

30.  If  the  sum  of  the  cubes  of  two  numbers  be  divided  by 

the  sum  of  the  numbers,  the  quotient  is  equal  to  the 
square  of  their  difference  increased  by  their  product. 
Express  this  algebraically. 

31.  Express  in  words  the  following  algebraic  statement: 

/^3 1^ 

=  (^  +  yj  —  ^y. 

x  —  y 


12  SUBSTITUTION. 


32.  The  square  on  the  diagonal  of  a  cube  is  equal  to  three 

times  the  square  on  the  edge.  Express  this  in  sym- 
bols, using  I  for  length  of  the  edge,  and  d  for  length 
of  the  diagonal. 

33.  Express  in  symbols  that  "  the  length  of  the  edge  of  the 

greatest  cube  that  can  be  cut  from  a  sphere  is  equal 
to  the  square  root  of  one-third  the  square  of  the 
diameter." 

34.  Express  in  symbols  that  any  ''rectangle  is  half  the 

rectangle  contained  by  the  diagonals  of  the  squares 
upon  two  adjacent  sides." 

The  square  on  the  diagonal  of  a  square  is  double  the  square 
on  a  side. 

35.  The  area  of  a  circle  is  equal  to  tt  times  the  square  of 

the  radius.  Express  this  in  symbols.  Also  express 
in  symbols  the  area  of  the  ring  between  two  con- 
centric circles. 

36.  The  volume  of  a  cylinder  is  equal  to  the  product  of  its 

height  into  the  area  of  the  base ;  that  of  a  cone  is 
one-third  of  this ;  and  that  of  a  sphere  is  two-thirds 
of  the  volume  of  the  circumscribing  cylinder.  Ex- 
press these  facts  in  symbols,  using  h  for  the  height 
of  the  cylinder,  and  r  for  the  radius  of  its  base. 


Ex.  4. 

Perform  the  additions  in  the  following  cases : 

1.  (h  —  a)x  ~\-  (c—  l>)y  and  {a  -f  l))x-\-{b  -f-  c)y. 

2.  ax~hy,  (a  —  h)x  —  (a  +  h)y,  and  (a-\-h)x  —  (h  —  a)y. 

3.  (7j--z)a'+(z  —  x)ah  +  (x-y)h^ 

and  (x  —  y)  a"^  —  (z  —  y)  ah  —  (x  ^-  z)  U^. 


FUNDAMENTAL    FORMULAS.  13 

4.  ax  +  ^y  +  cz,  hx-{-  cy  +  az,  and  ex  -\-  ay  -\-  hz. 

5.  {a  +  h)x'  +  (h+c)f  +  {a  +  c)z\  {h  +  c)x'  +  {a  +  c)f 

+  (a  +  h)z\  (a  +  c)x^  +  (ct  +  h)f  +  (b  +  c)z\  and 
-(^a  +  b  +  c){x'  +  f  +  z% 

6.  x{a-hy  +  y(h-^cf  +  z{c-a)\  y{a-hy  +  z(h-cf 

+  x{c  —  ay,  and  z{a  —  l)f  +  x(h  —  cf  +  y{c  —  af. 

7.  {a--h)x''  +  {h  -  c)y''  +  {c-  a)z\  {h-c)x^  +  {c-a)y'' 

+  {a-h)z\  and  {c  -  a)x'+{a-h)f+{b-c)z\ 

8.  (a  + ^)^  + (^  +  ^)3/ —  (c  + a)2;,  (^  +  c)  2; +  (c  +  a):25 

—  {a  +  h)y,  and  (a  +  c)3/ +  (a  +  ^)2;  —  (6  +  <?)a:. 

9.  a^-3a^-if5^    2h''-lb^  +  c\   ah-\b''  +  h\    and 

10.  ax^'  —  ^bx'',  —9ax''-\-1bx'',  and  —  85a;'*+ 10arr^ 

11.  What  will  {ax—by-\-cz)+{bx-\-cy  —  az)—{cx+ay+bz) 

become  when  a;— 3/— 2  —  1? 

Fundamental  Formulas  and  their  Application. 

By  Multiplication  we  obtain 

{x  +  o^){x  +  s)=^x''\-{t  +  s)x  +  o^s  [A] 

{x  +  r){x  +  s){x  +  t) 

-=x^  +  {r  +  s+t)x'  +  {rs  +  st+  tr)x  +  rst  [B] 

From  [A]  we  obtain  immediately 

{x  ±iyy  =  oc^  ^2xy -{-y'^  [1] 

{x  +  y+zy  =  x'  +  2xy  +  2xz  +  y^  +  2yz  +  z"  [2] 

{%af  =  %a}  +  2%ab  [3] 

{x  -{-y){x-y)  =  x'  —  f  [4] 
The  symbol  2  means  "  the  sum  of  all  such  terms  as." 


14  FUNDAMENTAL    FORMULAS. 

From  [B]  we  derive 

(x  zh  yY  =  x^  zhSx'^i/  -^3  xif  ±  y^  [5] 

=^  x^  ^  y^  ±iZxy  {x  -±1  y')  [6] 

{x-^y^zf  =  o^  +  f  +  z^ 

+  Zx\y+z)  +  ^y\z  +  x)  +  Zz\x  +  y) 
+  ^xyz  [7] 

^j^  +  f  +  z'  +  Z{x  +  y){:y  +  z){x  +  z)  [8] 

=^a^+f  +  z''  +  ?>{x  +  y  +  z){xy  +  yz  +  xz)  —  '^xyz    [9J 

{taf  =  Sa^  +  S^a:'b  +  6:$abc  [10] 

Formula  [1].     Examples. 

1.  We  have  at  once  (x  +  yy  -{-  {x  —  yy  =  2(x^  -{-  y^)  and 

(x  +  yy-{x-yy  =  4:xy. 

2.  (a-\-h  -{-  c~{-  dy  +  (a  —  b  —  c-{-  dy    may    be    written 

[(a  +  6^)  +  (^  +  ^)  J^  +  [(a  +  J)  -  (Z>  +  c)7, 

which  (Exam.  1) 

=  2[(a+c?7  +  (5  +  e)^]; 
similarly, 
(a-  Z>  +  c  -  cZ)'^  +  (a  +  5  -  ^  -  c^y 

=  l(^a-d)-{}>-c)J^\{a-d)-\-{b-c)J 

-2[(a-cZ7  +  (5-cn 

-\-{a-\-b-~c-dy 

=  2\{a^dy^q,+cy-\-{a-dy-\-{b-cy\ 

by  Exam.  1, 

=  ^{a'-\-b^-\-c^-\-d''). 


FUNDAMENTAL    FORMULAS.  15 

3.  Simplify  (^a  +  h+  cf  —  2{a  +  h  +  c)c  +  c\ 

This  is  the  square  of  a  binomial  of  which  the  first  term 
is  {a  +  h  +  c),  and  the  second  —c.     Hence  it  equals 

[(a  +  h  +  c)  -  cj  =  (a  +  h)\ 

4.  Simplify  {a  +  }))'- 2  (p?  +  V)  (a  +  hy  +  2 (a'  +  b'). 
By  Exam.  1,  2(a'  +  b')  =  (a^  +  bj  +  (a'  -  by. 
Hence,  the  given  expression  equals 

(a  +  by-2  (a'  +  b')  {a  +  bf  +  (a'  +  by  +  (a'  -  by 
=  [(a  +  by  -  (a'  +  b')J  +  (a'  -  by 
=  a'  +  2  a'  b'  +  b'^  (a'  +  by. 


Simplify  :  ^^-  ^• 

1.  (:^+3y7  +  (rr-3y2)^;   (^a' + 'dby  ~  (^a' -  Sby. 
Show  that : 

2.  (mx  +  riT/y  +  (nx  —  Tn^y  =  (rn^  +  ^^)  (x^  +  'if). 

3.  (mx  —  nyy  —  (n:r  —  rnyy  =  (m^  —  n'^)  (x"^  —  y^). 
Simplify  : 

4.  [(a  +  3S)^  +  2(a+Sb)(a  -b)  +  {a~  by]  (a  -  by 

5.  (.T  +  3)' +  (:?;  +  4)^  -  (^r  +  57, 

and  (i^^^-2y0^-(i3/^  +  2a;7. 

6.  {a  +  b  +  cy  +  (b  +  cy  ~  2{b  -\.c){a+b  +  cy 
Show  that : 

7 .  {ax  +  byy  +  {ex  +  c^)^  +  {ay  -  bxy  +  (^y  -  dxy 

=  {a'  +  b'  +  c\+  d')  {x'  +  y'). 

Simplify : 

8.  {x-Syy  +  {Sx'~yy-2{Sx'-y){x-^y 


16  FUNDAMENTAL    FORMULAS. 

9.  \x^  -\-xy  —  y^y  —  {x^  —  xy  —  y^)^ 

(1  +  2:r  +  4:?;7+  {\-1x-\-  ^x^)\ 

10.  If  a  +  Z>  —  —  \c,  show  that 

+  2(2Z)-c)(2c-a)  +  2(2c-a)(2a-&)  =  xV^'- 
Simplify : 

11.  2(a  -  ^)/  -  (a  -  2Z))^  (a^  +  \ab  +  Z^^)^  -  (a^  +  Z)^. 

12.  (a  +  Z))2-(5  +  c)^  +  (c  +  c^)^-(c^+a)l 

13.  {\x-yJ-\-{\y-zJ-^{\z-xJ-^2i^\x-y){\z-x) 

Show  that : 

14.  (rr  — yy  +  (y-2)'  +  (2;-rr)^ 

-2(r.-y)(.-y)  +  2(y-rr)(0-^)+2(^-y)(^-:r). 

Simplify : 

15.  (1  +  xy  -  2(1  +  x^){^.  +  xj  +  2(1  +  ^0- 

16.  {x^y-\-zy-{x-^y-zy~{y  +  z-xy-{z-^x-y)\ 

17.  (:r-2y  +  30)^+(3^-2y)^  +  2(rir-23/  +  3z)(2y-3^). 

18.  (aH^'--c7  +  (^'-^?  +  2(Z^'-c^)(aH^'-0. 

19.  (r,  +  y)^  +  (^_y)^_2(a;-y)^(:r  +  y)l 

20.  (5a  +  35)'^+16(3a  +  5)^-(13a  +  5Z))l     . 

Show  that : 

21.  (3a-Z^)^  +  (3Z)-c)^  +  (3c-a)^-2(Z)-3a)(3Z)-c) 

+  2(3Z>-c)(3c-a)-2(a-3c)(3a-Z^) 
— 4(a  +  Z>+6?)'=:0. 

22.  If  z"-  =  2xy,  show  that  (2a;'  —  y^  +  (2;'  —  2y''y 

+  (a:'  -  2zy  -  2(2^;'^  -  y') (z'  -  2^^^) 

+  2  (a;2  -  2^')  (z'  -  2/)  -  2  (a;'  -  2  2')  (2x'  ~  f) 


FUNDAMENTAL    FORMULAS.  17 

Simplify 

23.  (l  +  x  +  x'  +  a^y  +  (l-~x-x'  +  xy 

+  (1  -  ^^  +  ^-^  -  ^y  +  (l+x~x'-- xj. 

24 .  {ax  +  hyy  -  2  (a'  x'  +  h'  y')  (a:t^  +  hyj  +  2  (a*  x'  +  h'  y') . 

Formulas  [2]  and  [3].     Examples. 

1.  (1-2:^  +  3:^7  =  1-4:^;+    6:r' 

+    4:^^^~12:r^ 

Ar^x' 

=  1  -  4:^;  +  10:^'^  -  12r^  +  ^x' 

2.  {ph-^hc^cay 

=  a'b'  +  2ab'c  +  2a'bc  +  b'c'  +  2abc'  +  c'a' 
=  a'b'  +  Pc'  +  c'd  +  2abc{a  -\-b-\-c). 

3.  [(•^  +  y)^  +  ^^  +  yT 

^{x-\-yy  +  2{x-\-  yjix'  +  y^)  +  :^;^  +  2x'f  +  y^ 

=  (^  +  y)*+  (^  +  2/)'  [(^  +  yy  +  (^  -  y)'1 

+  :r^+2:^Y  +  / 
-  2(:^;  +  y)^  +  {x^  -  yj  +  x' +  2x'f  +  / 
-2[(a;  +  yy  +  a;^  +  y*]. 

4.  (a;'  +  :ry  +  3/7 

^x'  +  2x\j  +  2x^y'  +  :r'y'  +  2:^?/'  +  i/ 

=^{^  +  y)'^'  +  ^'y'  +  y\^  +  yj- 

5.  In  Exam.  3,  substitute  b  —  c  for  x,  c  —  a  for  ?/,  and  con- 

sequently  b  —  a  for  ^+3/;    then,   since   (b  —  of  =^ 
(a— by,  Exam.  3  gives 
l^a-by  +  (b-cy  +  (c-a)J 
=  2  [(a-  by  +  (b-  cy  +(c-  ay] . 

6.  Making  tlie  same  substitutions  in  Exam.  4,  we  have 
(a'  +  b'  +  c'~ab-bc-  cay 

=  {a-by(b-cy+(b-cy(c-ay+(c-ay(a-by^ 


18  FUNDAMENTAL    FORMULAS. 

or,  multiplying  both  sides  by  4, 

-=  4  (a  -  hy  (b  -cf  +  A  (h-cj  {c-af+^  {c-af  {a-h)\ 
Hence,  from  Exam.  5, 

{a--hy  +  {b-cy+{c-ay 

=  2  {a-bj  (b-cf + 2  {b-cy  {c-af  +  2  {c-a)\a-  b)\ 

Expand:  ^^-  ^• 

1.  (1-2^+3^^-4^7;  (l-x  +  x'-xj. 

2.  {l-2x  +  2x'-?>x^-~xj]  {l  +  '^x  +  ^x'^  +  x^. 

3.  \2a-b-c'~iy',  (l-x  +  y  +  zy^  (i^_i2/+6^)l 

4 .  (x^  ~x^y  +  xy^  —  y^y  ;  (ax  +  bx^  +  c?^^  +  (i:^;*)^ 

5 .  Show  that  (a'  +  b''  +  c%x^  +  y'  +  2')  -  {ax  +  by  +  czf 

=  (o^y  —  bxy  4-  (<?^  —  azy  +  (^2;  —  ^y)^ 

6.  Show  that  (a  -\-  b)  X  -\-  (b  -\-  c)y  +  {c  -\~  a)  z  multiplied 

by  (a  —  b) X  -\-  {b  ~  c)y  -{-  (c  —  a) z  is  equal  to  the 
difference  of  the  squares  of  two  trinomials. 

7.  Showthat(a— 5)(a— c)  +  (Z)  — c?)(5  — a)  +  (c— a)((?— 5) 

-  i[(a  -  by  +  {b-  cy  +  {c~  ay]  =  o. 

8.  Simplify  [a--(b-c)Y+[b-(c-a)y+[c-(a-b)y. 

9.  Show  that  (a''+b'-xy+(ai'+b,'-xj+2(aa,  +  bb,y 

=  (a'  +  a,'  -  xj  +  (b'  +  5i^  -  xy  +  2(ab  +  a,  by. 

10.  Show  that  [(a-b)  (b-c)+(b-c)  (c-a)  +  (c~a)  (a-b)] ' 

=  {a-by{b-cy+(b~cy(c-ay+{c-ay{a-by. 

11.  ^c\\xd.i:Q2a  —  ^bx~\cx-\-2dx. 

12.  If  rj;  +  y  +  2;  =  0,  show  that 

x'  +  y'  +  z'  =  (x'  -  yO'  +  (f  -  ^y  +  (^'  -  ^7' 

13.  Show  that  a'  (b  +  c?)^  +  Z^^ (^  +  a)'  +  c\a  +  by 

+  2a5c(a  +  ^  +  c)  =  2(ab  +  Z^^  +  m)l 


FUNDAMENTAL    FORMULAS.  19 

§  5.  To  apply  formula  [4]  to  obtain  the  product  of  two 
factors  which  differ  only  in  the  signs  of  some  of  their  terms, 
group  together  all  the  terms  whose  signs  are  the  same  in 
one  factor  as  they  are  in  the  other,  and  then  form  into  a 
second  group  all  the  other  terms. 

1 .    Multiply  a-i-h  —  c-{~dhj  a  —  h  —  c  —  d. 

Here  the  first  group  is  a—c^  the  second  h-{-d.      Hence, 

we  have 
[(a-c)  +  (b  +  d)][(a-c)-(b  +  d)] 
=  (a-cy-(b  +  d)\ 

^  [(1  +  Sx')  +  (Sx  +  x')]  [(1  +3;r^)  -  (3x  +  x')] 
^-ll  +  Sxy-iSx  +  a^y 
=:l~Sx^  +  Sx'-x\ 

3.  Find  the  continued  product   of   a-\-b-{-c,    b-}-c  —  a, 

c-\-a  —  b,  and  a-j-b  -  c. 
The  first  pair  of  factors  gives  [(b -\- c) -j- a]  [(b  -j-c)  —  a] 
=  (b  +  cy  -  0?=:  b''  +  2bc  +  c'  -  a\ 

The  second  pair  gives  \_a  —  {b  —  c)'\  \_a  +  (b  —  c)] 

=  o^~~b''  +  2bc-~c\ 
The  only  term  whose  sign  is  the  same  in  both  these 

results  is  2bc\  hence,  grouping  the  other  terms,  we 

have 
[2  be  +  (b'  +  c'-  a')]  [2  be  -  (b'  +  c'-  a')] 

=  {2bcy-(b''  +  c^-a?y 

--=  2o?b~  +  2b'' c"  +  20" o? -  a' -¥- c\ 

4.  Show  that  (a^  -\-ab-\-  by  -  c(^  ^  =  (a^  +  aby  +  {ab  +  h'J. 
The  expression  =^  (a"  +  b'')  (a"  +  2ab  +  b') 

=  (a'  +  b')  (a  +  by 

=-  a\a  +  by  +  b''  {a+by 

=  (a'  +  aby  +  (ab  +  bj. 


20  FUNDAMENTxlL    FORMULAS. 

Ex.  7. 

1.  {o?  +  2ah-\-h'')(o?-2ah  +  h''). 

2.  {^x'-xy  +  y''){^x'  +  y''  +  xy). 

3.  (o?-ah  +  2h''){a'  +  ah  +  2h'')]    {x' +  4:xy){x' -^xy). 

4-  [(^  +  y) ^  -  y  (^  -  y)]  [(^  ~y)^-y{y-  ^)\ 

5.  Simplify  (ri;+3)(^-3)  +  (^+4)(:?;-4)-(:r+5)(:?;-5). 

6.  Simplify  (1  +  xy  +  (1  -  ^)*  -  2  (1  -  xj, 

7 .  (:r^  +  y'^y  -  (2  ^y)^  -  (x^  -  7/^)1 

8.  (2a^-3Z^^  +  4c0(2«'  +  3Z^^-4c^). 

9.  (2a+Z>-3c)(/^  +  3c-2a);  (2a-Z^-3c)(5-3c--2a). 

10.  (^^  +  /)(^^'  +  y')(^  +  3/)(^-y)- 

11.  (^'  +  ^y  +  2/')  (^'  —  xy  +  y')  (^*  -  ^r'y"  +  i/). 

12.  (a  +  ^-aZ)-l)(a  +  Z>  +  a^  +  l). 

13.  If  a'  =  Z^'  +  c*,  show  tliat 

(a^  +  Z^^  +  c^)  (Z>^  +  C'-  a')  (c'  +  o^~  b')  (a'  +  b'  -  c') 

Simplify : 

14.  (.^'  +  y'-|^y)(^'  +  y'  +  |^y). 

15.  (x' -^x^+^x'^-^x+l) {x'  +  2x^  +  ?>x^  +  2x  +  1). 

16.  Multiply  {2x  —  y)  o?  —  (x -{-  y)  ax  +  2;^  by  (2^  —  y)  a^ 

-\-(x  ■^y)ax  —  x^. 

Show  that : 

17 .  {d'  +  b''  +  c^  +  ab+bc  +  caf  -  {ab -^bc-\-  caf 

=^{a-\-b-Y  6f  {a!'  ^b^  -\-  &), 

18.  (a'  +  Z^2  +  c^  +  ab-\-bc-\-  caf  ~  {ci^  +  ah  +  ca-  bcf 

=  [(a  +  b)(b  +  c)y  +  [(b  +  c)(c  +  a)]\ 

19.  4  (ab  +  cc^)'  -  (a^  +  b'-c'-  dj 

=  (a+b+c-d)  (a+b-c+d)  {c+d+a-b)  {c+d-a+b). 


FUNDAMENTAL    FORMULAS.  21 

20.  Find  the  product  of :   x^  -\-y^  -\-z^ —'Ixy  ^  2xz  —  2yz 

and  :r^  +  y^  +  2;^  —  2^^y  —  2xz  +  27/z. 

21.  (x'  +  y'  +  xy^2)(x'-xi/-^2+f)(x'-y'). 

22.  (l-6a  +  9a')(i  +  2a  +  3a2). 

23.  [(771 -j- n) -\~  (p -{- q)](m  — q-^-p  —  n). 

24.  1  +  :z;  +  ^^  ^^  +  ^  —  1,  ^^  —  ^  +  1,  and  l-\-x  —  x"^. 

25 .  (a  -  ^^)2  (a  +  5^)^  (a^  +  bj  (a'  +  5«)l 

26.  Show  that  (x"^  +  xy  +  y^  {x^  ~  ^y  +  y'^J'  —  ipc^iTf 

=  {x'  +  x'yy  +  {x'f  +  yy. 

Formula  A.     Examples. 

1.  Multiply  x^  —  x  +  bhj  x^  —  x-~l. 

Here  the  common  term  \^  x"^  —  X]  the  other  terms,  ~\-b 
and  —7.     Hence,  the  product  equals 

{x'-xy  +  {~1  +  b){x'-x)  +  {-1xb) 
=  (x'  -  xy  -2(x'-  x)  -  35 
=  x''~2x^-x^^2x-Zb, 

2.  {x  —  a)  (^  —  3a)  (r^  +  4a)  {x  +  6a). 

Taking  the  first  and  third  factors  together,  and  the 
second  and  fourth,  we  have  the  product  equals 

{x^  +  3a^  -  4a')  {x?  -{-Zax-  18a') 

=  (a;'+3a:r)'-(4a'  +  18a')(:^'  +  3a:r)-72a^etc. 

Ex.  8. 

Find  the  products  of : 

1.  {x^  -\-2x^Z){x^  -\-2x-  ^)\  {x-y-\-^z)(x~y-\-hz). 

2.  (^+l)(^+5)(:r+2)(a;+4);  {x'-\-a--h){x'^2h-d). 

3.  (a'-3)(a'-l)(a'  +  5)(a'  +  7);  (:rH^'+l)(:r*+a;'-2). 


22  FUNDAMENTAL    FORMULAS. 

4.  [(x  +  yj  -  ^xy-\  \{x  +  yy  +  hxy\ 

5.  (a:»  +  a+7)(x"-a-9);    j'^  +  l- l")  g  +  |  + 3 

6.  (wa;  +  y  +  3)(«a;  +  2/  +  7). 

7.  (x  +  a  — y)(a-+a  +  3y). 

8.  (a;'"  +  «''-a)(ar=»  +  ar"-^»). 

9.  (i,s*_y^  +  2)aa;*-y^-4). 

11.  rr-2+V2,  :i;-2+V3,  :t'-2-V2,  .r-2-V3. 

12.  (:r  +  a  +  Z>)  (^  +  ^  —  c)  (:i:  —  a  +  5)  (:?;  +  ^  +  c?). 

13.  ia-^l  -^  c){cL-\-h  -^  d)-\'{a-\-  c  -^  d){h  -^  c  -^  d) 

~{p.-\-h^c-\-d)\ 

14.  Show  that  (2a +  25-6?) (2Z>  + 2c -a) 

+  (2  c+2  a-5)  (2  a+2  &-c)  +  (2  ^>+  2  c-a)  (2  6^+2  a-5) 
=  9(a&  +  5c+<?a). 

Formulas  [5]  and  [6].     Examples. 

1.  We  get  at  once 

(^  +  yj  +  C^  -  yf  =  ^^(^'  +  3y^) ; 
(^  +  yf  -  C^  -  yf  =  2y(^^'  +  f)- 

2.  Simplify  (a+b  +  cy-S(a+b+cyc+S(a+h+c)c'-c\ 
This  comes  under  formula  [5],  the  first  term  being 

a-\-b-{-  c]  the  second,  —c.     Hence,  the  expression  is 
[(a  +  b  +  c)-cY  =  (a  +  bf. 

3 .  Show  that  (x^  +  ^y  +  y^  +  (x7j  —  ^'^  —  ff 

-6x7/(x'  +  x'^7/  +  y')  =  Sx'7/\ 
This  comes  under  formula   [6],  the  first  term  being 
(x^  +  xy  +  'if),  and  the  second  —  {x^  —  xy  +  if)  ;   we 
have,  therefore, 
[(^  +  a;y  +  f)  -  {o?  -  xy -\-  f)Y  =  {^xy)'  =  ^o^f. 


FUNDAMENTAL    FOEMULAS.  23 

Simplify:  ^^-  ^• 

1.  (i-xj  +  (i  +  xy;  (x'  +  x7/y-(x'-x7/y. 

2.  (a+2bf-(a~by',  (Sa-hf  -  (3a~2by. 

3.  (x  +  y  -  zf  +  S(x  +  y  -  zy  z  +  z'  +  ^(x  +  y  -  z)z'. 

4.  (a-hy  +  (a  +  by  +  6a(a'-b'). 

5.  (^-^y  +  (a:  +  yy  +  S(x-7/y(x+y)-S(i/-xXx  +  7jy. 

6.  (l  +  ^  +  ^7'-(l-^  +  ^7'-6^(l+a;^  +  ;rO. 

7.  (a-b~cy  +  (b  +  cy  +  d(ib  +  cy{a-b-c) 

+  3(a-b-cy(b  +  cy 

8.  (3^-4//+52;7-(52;-4y7  +  3(52;-4y)X3a;-4y  +  5z) 

-  3  (3x -  4y  +  52;)2  (5z  -  4y). 

9.  (1 +  ^  +  :^2)3_j_  3(1  _^>)^2  +  :i^^)  + (1-^)1 

Show  that : 

10.  a(a-2by-b(b--2ay  =  (a-b)(a  +  by. 

11.  a' (a' -2  bj  +  5-^  (2  a'  -  Z>')'  -  (a'  -  b')  (a'  +  bj. 

Simplify : 

12.  (x'+x7/+yy+6(x'+y')(x'  +  x2/+7/)  +  (x'-X2/+yJ, 

I         Show  that : 

13.  a^ (a^  +  2 bj  +  F (2 a^  +  bj  +  (3 a^ bj 
,  ={a^+1a'¥  +  bj. 

*         Simplify : 

14 .  (ax  +  byy  -f  ^-^  y^  +  ^^  ^^  —  3  aZ>:ry  (ax  +  ^y) . 

15.  What  will  a^  -\- h^  -{-  c^  —  Sabc  become  when 
I  a  +  b+c  =  0. 

16.  Find  the  value  of  r^:^  — /  + /  + 3^^3/^2;^  when 

i 


24  fundamental  formulas. 

Formulas  [7],  [8],  and  [9].     Examples. 

1.  Simplify  (2x-?>yy  +  {^y-~bxy+{?>x-yy 

-?>{2x-  3y) (4y  -bx){^x- y). 
By  [9]  this  is  seen  to  be 

[{2x-?>y)  +  {^y-bx)  +  (?>x-y)J^{0y  =  0. 

2.  Prove  that  {a  -  hf  +  (b  -  cf  +  (c  -  af 

=  S(a-b)(b-c)(c-a). 
In  [9]  substitute  a  — b  for  x,  b  —  c  for  y,  and  c~a  for  z; 
for  these  values  x -\- y -{- z  =  0,  and  the  identity  ap- 
pears at  once. 

3.  TvoYe (a+b  +  cy-(b  +  c-ay~(a  +  c-by-(a  +  b-cf 

=  24:abc. 
In  [8]  let  X  =  b-\-  c  —  a,  y  =  c-\-  a  —  b,  z=^a-{-b  —  c', 
and  therefore,  x-\-y  =  2c,  y-}- z  =  2a,  z-{-x=^2b] 
and  this  identity  at  once  appears. 

Ex.  10. 

1.  Cube  the  following : 

l-x  +  x';  a-b-c]  l-2x +  Sx^ -4:x\ 

Simplify : 

2.  (x''  +  2x-iy  +  (2x-l)(x'  +  2x-2)-(x'  +  Sx'-l)\ 

Prove  that : 

3.  xyz  +  (x  +  y)(y  +  z)(z  +  x)  =  (x  +  y-\-z)(xy+yz  +  zx)^ 

4.  (ax  —  byY  -\-  a^y^  —  ¥  x^  +  3  abxy  (ax  —  by) 

=  (a^-b')(x^  +  f). 

*  Note  that  the  right-hand  member  is  formed  from  the  left-hand  one  by 
changing  additions  into  multiplications,  and  multiplications  into  additions ; 
hence,  in  (a;  -f  y  -f  2)  X  (a:  X  2/  +  2/  X  z  +  z  X  a?)  the  signs  -f  and  x  may  be 
interchanged  throughout  without  altering  the  value  of  the  expression. 


FUNDAMENTAL    FORMULAS.  25 

Simplify  : 

+  ?>{x-y-2z){y-z-2x)(z-x-2ij). 

6.  {2x'  -  3y^  +  4^7  +  {2^  -  ^z'  +  40;^ 

7.  {2ax-hyy  +  (2hy-czy  +  {2cz-axy 

-\~S(2ax-\-hy  —  cz)  (2  hy-{~cz  —  ax)  (2cz  +  arr  —  by). 

Prove : 

8.  (a^  +  Sx'y-y^y+l^xyix  +  ij)]' 

=.[(x-yy  +  9x'  y]  (x^  +  xy  +  y'f. 

9.  9(x'  +  f  +  z')-(x  +  y  +  zy=(4:x+4:y  +  z)(x-yy 

+  (^y  +  4:z  +  x)(y-zy  +  {4:z  +  4:x  +  y){z-x)\ 

10.  If  x-\-y-{-z  =  0,  show  tliat  x^  +  y^  +  2;^  —  Srrys;. 

11.  Ux  =  2y+^z,  Bhow  thsit  x^ -Sy" -27 z' -  lSxyz  =  0. 
Show  that : 

12.  (x'+xy+7ff+(x'-xy+y''y  +  8z^-(jz\x'+xy+y') 

Prove  that : 

13.  8(a  +  b  +  cy-(a+by-(b  +  cj  -{c  +  af 

=  ^(2a  +  b  +  c){a  +  2b  +  c){cL  +  b  +  2c). 

Prove  the  following  : 

14.  (ax  —  byy  +  ^^ y^  =  a^ :z;^  +  3  abxy  (by  —  ax). 

15.  a'^  +  b^  +  c^-^abc 

r=^[{a-by  +  (b-cy  +  {c-ay](a  +  b  +  c). 

16.  (a  +  b+c)[(a  +  b  —  c)(b  +  c  —  a)  +  {b  +  c  —  a){c+a  —  b) 

+  (c  +  a-b)(a  +  b-c)] 

=^(a-]-b  —c)(b-\-  c~  a)(c-{-a  —  b) -\-8abc. 

17.  (a  +  b  +  cy-?>[a(b-  cy  +  b(c  ~ay+c  (a  -  by] 

=^a^  +  b'  +  c^  +  24:abc. 


26  FUNDAMENTAL    FORMULAS. 

18.  {a  +  h  +  1  c){a-hy  +  {b  +  c  +1  a)ih  -  cf 

+  {c  +  a+1h){c~ay=:^2{a  +  h  +  cf  -  bA^ahc. 

19.  {a  +  h  +  c)[(2a--b)(2h~c)  +  (2h-c){2c-a) 

+  (2c~a)(2a-h)]=^(2a-h){2h-c){2c-a) 
+  {201  +  h -  c){2h  +  c  -  a)  {2c  +  a-  h). 

20.  If  x^  {y -{- z)  =^  a^ ,    qf  (z -{- x)  =  If ,   z^  (x -{- y)  =  c^ ,   and 

xyz  =  abc,  show  that 

a'+P  +  c^  +  2abc  =  (x  +  7j)(y  +  z)(z  +  x). 

y  Expansion  of  Binomials. 

We  have,  from  formula  [5], 

(a+bf  =  a'  +  Sa'b  +  Sab'  +  P', 
multiplying  by  a-]-b,  we  obtain 

(a  +  by  =  a'  +  Aa'b  +  6a''b''  +  4:aP+b'',    ■ 
multiplying  this  by  a-\-b,  we  obtain 

{a  +  bf  =  o^-\-ba'b  +  lOa^b''+lOo?b^  +  bab'  +  bK 

From  these  examples  we  derive  the  following  law  for  the 
formation  of  the  terms  in  the  expansion  of  a-\-b  to  any 
required  power : 

I.  The  exponent  of  a,  in  the  first  term,  is  that  of  the  given 
power,  and  decreases  by  unity  in  each  succeeding  term ;  the 
exponent  of  b  begins  with  unity  in  the  second  term,  and 
increases  by  unity  in  each  succeeding  term. 

II.  The  coefficient  of  the  first  term  is  unity,  and  the  co- 
efficient of  any  other  term  is  found  by  multiplying  the 
coefficient  of  the  preceding  term  by  the  exponent  of  a  in 
that  term,  and  dividing  the  product  by  the  number  of  that 
preceding  term. 

It  will  be  observed  that  the  coefficients  equally  distant 
from  the  extremes  of  the  expansion  are  equal. 


MULTIPLICATION   AND   DIVISION. 


'Zi 


1. 
2. 

3. 

4. 
5. 


8. 
9. 


Ex.  11. 

Expand  (x  +  yf ;  {x  +  yj  ;  {x  +  yf  ;  (x  +  yf. 

What  will  be  the  law  of  signs,  if  ~y  be  written  for  y 
in[l]? 

Expand  (a  -  hf  ;  (a  -  2  Z^)* ;  (2  Z^  -  a)*- 

Expand  (1  +  mf  ;  (m  +  1/  ;  (2  m  +  1/. 

What  is  the  coefficient  of  the  fourth  term  in  (a  —  hy^  ? 

Expand  (x'  -y)'-  (a  -  2  hj  ;  {a' -2  bj. 

In  the  expansion  of  (a  —  by^  the  third  term  is  66  a^^  b^ ; 
find  the  fifth  and  sixth  terms. 

Show  that  (x  +  yf  —  x^  —  y^  =  5  xy  (x  +  y)  (x^  +  ^y  +  y^)- 

From  [8]  show  that  2  [(a  -  bf  +  (b  —  cf  +  (c  —  af] 


(b-cf 


Hounee's  Methods  of  Multiplication  and  Division. 

Examples. 

1 .    Find  the  product  of  kx^  +  Ix^  +  mx  +  n  and  a:i'^  +  5:?;  +  <^- 
Write  the  multiplier  in  a  column  to  the  left  of  the  mul- 
tiplicand, placing  each  term  in  the  same  horizontal 
line  with  the  partial  product  it  gives : 
Jcx^   -\-lx^     -\-mx       -\-n  Q 


ax 

+  bx 


akx"   -\'alx^   -\-amx^    -{-ana^  pi 

-\~bkx^  -\-blx^       -\-bmx'^-{-bnx  p^ 

-\-cJcx^      -\'clx'^  -\-cmx-\-C7i ps 


a'koi^-[-{cil'\-b'k)  x^-[-(am-\-bl-\-c'k)  x^-\-(an-{-bm-\-cl)  x^ 

+(Z>n+cm).r+C7i P 


28 


MULTIPLICATION    AND   DIVISION. 


§  6.  The  above  example  has  been  given  in  full,  the  pow- 
ers of  X  being  inserted ;  in  the  following  example  detached 
coefficients  are  used.  It  is  evident  that,  if  the  coefficient 
of  the  first  term  of  the  multiplier  be  unity,  the  coefficients 
of  the  multiplicand  will  be  the  same  as  those  of  the  first 
partial  product,  and  may  be  used  for  them,  thus  saving  the 
repetition  of  a  line. 

2.    Multiply  ?>x'  —  2af  —  2rr  +  3  by  x"  +  3r^  —  2. 


1 
+  3 

-2 


3-2+0-2+3 

+  9-6+0-6+9 
_    6     +4     -0     +   4 


3^6 _j_  7^5 _  Y^^'^ -\-2x^-Zx^-\-\?>x~^ 
3.    Find  the  product  of  {x  —  3)  {x  +  4)  {x  —  2)  (:r  —  5). 

+  4 

-2 

-5 


4.    Multiply  ^'  — 4r^' +  2^  — 3  by  2^'  — 3. 

1     -4     +2     -3 

{x^  Vio?^  x^) 
0 
0         0 

+  12     -6    +9 


1   -3 

+  4     - 12 

1+1     -12 

-2     -   2     +24 

1-1     -14     +24 
-5     +5     +70    - 

-120 

x*-Gx'-   9a;'  +  94x- 

-120 

2 

2 

-8     +4 

-6 

0 

0         0 

0 

0 

0 

0 

3 

-3 

2x^  —  S:x^  +  ^x'  -  9x^  +  12^;''  —  6a;  +  9 
In  this  example  the  missing  terms  of  the  multiplier  are 


MULTIPLICATION    AND    DIVISION. 


29 


supplied  by  zeros ;  but,  instead  of  writing  the  zeros  as  in 
the  example,  we  may,  as  in  ordinary  arithmetical  multipli- 
cation, ''skip  a  line"  for  every  missing  term. 

5 .    Multiply  x'  -^x'  +  lhj  x'-x'  +  ?>. 


1 
-1 

+  3 


1  +  0-2  +  0+1  {x'xx'  =  x^) 

-1-0+2-0-1 

+3+0-6+0+3 


-?>x^     +^x^ 


+  3 


6.    Find  the  value  of 

{x  +  2)  (x  +  3)  (:r  +  4)  {x  +  5)  -  9  {x+2)  (^+3)  {x+  4) 
+  3(:^  +  2)(r^  +  3)  +  77  (:^:  +  2)  -  85. 


1  +5 
-9 

-16 
+   3 

+  4 

1  -4 

+  4 

+  3 

1  +0 
+  3 

-13 

+   0 

-39 

+  77 

+  2 

1  +3 
+  2 

-13 

+    6 

+  38 

-26    +76 
-85 

x'  +  ba?-    Ix 

•'+12a;-    9 

7.    Find  the  coefficient  of  x^  in  the  product  of 
x^  —  ax^  +  hx^  —  cx-{-  d  and  x"^  -^px  +  q. 


1     ~ a     ~\-b     ~  c    -\~  d 
~ap 

+  {b  —  ap  +  q) 

30  MULTIPLICATION    AND    DIVISION. 

Ex.  12. 

Find  the  product  of : 

1.  (1  +  X  +  x"  +  r^'  +  x') {\-x''-^x^-x'^-\-x^-x^^-\-x'''), 

2.  (1,+  a;')  (1  —  rr^  +  x^)  (1  +  ^  +  ^'  +  ^'  +  x"). 

3.  (ri— 5)(:r+G)(2:-7)(a;+8);   (2^^-a;'^+l)(:i;^-rr  +  2). 

4.  (:r^  +  5:^^  -  \^x-  \){x^  -  5x'  -  16:?:+  1). 

5.  {6x'  -  x'  +  2x'  -  2a^'"^  +  2^2+19^  +  6)  (Sx'+ix  +  l). 
Obtain  the  coefficients  of  x^  and  lower  powers  in 

6.  {^l.-f-^X       -gX  +yg-^        12  8"^  )\^        2"^       "8*^        T6^        TTS^  )' 

7.  Multiply  2a;^  -  x' +  Sx -Ahj  Sx^  -  2x^  -  x-1. 
Simplify  the  following : 

8.  (x+l)(x  +  2)(x  +  3)  +  S(x+l)(x+2)-10(x+l)  +  9. 

9.  x(x+  1) (x  +  2)(x  +  S)~Sx(x+  1) (:r  +  2) 

—  2:r(:^+l)  +  2^. 

10.  :^  (:^  -  1)  (:?;  -  2)  (^  -  3)  +  3  :r  (^'  -  1)  (^  -  2) 

-2ri'(^-l)-2^. 

11.  (x~l)(x  +  l)(x  +  S)(x  +  5)~14:(x-l)(x+l)  +  l. 

12.  Given  that  the  sum  of  the  four  following  factors  is  —1, 

find  (i.)  the  product  of  the  first  pair  ;  (ii.)  the  product 
of  the  second  pair  ;  and  (iii.)  the  product  of  the  sum 
of  the  first  pair  by  the  sum  of  the  second  pair : 
(i.)  X  +x'  +  x^^  +  x^\ 
(ii.)  x' .+  x^  +  x^  +  x^\ 
(iii.)  x^  +  x''  +  x'''  +  x'\ 
(iv.)  x^  +  x'  +  x^^'-^-x''. 

13.  Given  that  the  sum  of  the  three  following  factors  is 

equal  to  —1,  find  their  product : 

(i.)  X  +x^  +  x^  +x'\ 

(ii.)  x'  +  x^  +  x^^  +  x^\ 

(iii.)  x'  +  x^  +  x''  +x\ 


MULTIPLICATION    AND    DIVISION.  31 

§  7.  Were  it  required  to  divide  the  product  JP  in  the  first 
of  the  above  examples  by  ax^  -{-hx-]-  c,  it  is  evident  that 
could  we  find  and  subtract  from  P  the  partial  products  p^, 
p^  (or,  what  would  give  the  same  result,  could  we  add  them 
with  the  sign  of  each  term  changed),  there  would  remain 
the  partial  product  J9i,  which,  divided  by  the  monomial  ax'^, 
would  give  the  quotient  Q.  This  is  what  Horner's  method 
does,  the  change  of  sign  being  secured  by  changing  the 
signs  of  h  and  c,  which  are  factors  in  each  term  oi  p^,  p^, 
respectively. 


1. 

—hx 
—c 


ahx^^-{al-\-hh)x*'-\-{am^r  hl-\-  ck)x^ -\-{an-\-  bm-\-  cT)x^-^{hn-\-cm)x-\-cn..P 

—hkx^  —hlx?  —hmx^        —hnx p.^ 

— chx^  —  clx^  —  cmx  —  en .  .p^ 


ax 


aks^         -\-alx^  +amx^  ^anx"^ .^^ 


kx^         -\-lx^  -^mx  +n Q 

The  dividend  and  divisor  are  arranged  as  in  the  exam- 
ple, the  sign  of  every  term  in  the  divisor,  except  the  first, 
being  changed  in  order  to  turn  the  subtractions  into  addi- 
tions. The  first  term  of  the  dividend  (alcx^)  is  brought 
down  into  the  line  of  pi ;  dividing  this  by  ax'^,  the  first  term 
of  the  divisor,  we  get  kx^,  the  first  tervi  of  the  quotient. 
Multiplying  this  term  hx^  by  —hx  and  —  <?,  respectively, 
and  writing  the  products  in  the  proper  columns  and  rows^ 
makes  all  ready  to  give  the  second  term  of />i,  which  is  got 
by  simply  adding  up  the  second  column  of  the  work,  giving 
alx^.  Dividing  this  second  term  of  ^i  by  ax^  gives  Ix'^,  the 
second  term  of  the  quotient.  Multiply  Ix^  by  ~hx  and  —  <?, 
respectively,  and  proceed  in  the  same  way  as  was  done  in 
getting  the  second  term  of  the  quotient,  and  the  third  will 
be  obtained.  Repeating  the  steps,  the  complete  quotient 
and  the  remainder  will  finally  be  obtained. 

Should  the  coefiicient  of  the  first  term  of  the  divisor  be 
unity ^  the  coefficients  of  the  line    Q  will  be  the  same  as 


32 


MULTIPLICATION    AND    DIVISION. 


those  of  pi,  and  the  line 
one  line  does  for  both. 


need  not  be  written  down,  since 


3 

+  7 

— 

12  +  2 

-3  +  13- 

-6 

-9 

+ 

6- 

-0 

+  6- 

9 

+ 

6- 

-4 

+  0- 

4  +  6 

Sx' 

^-2^  + 

0- 

-2^  +  3 

2.  Divide  dx'+7x^-12x'+2x'-Sx'  +  lSx-6  by  r^'+3^-2. 

(x^ -^  x"^  ^=  x^) 
-3 
+  2 

Compare  this  example  with  the  second  example  of  Horn- 
er's Multiplication,  performing  a  step  in  multiplication, 
then  the  corresponding  step  in  division ;  then  another  step 
in  multiplication,  and  the  second  (corresponding)  step  in 
division  ;  and  so  on. 

3.  Divide  rr^-3:^«+4:^*+18^^-7^+12  by  :^'-3^^'+3:r-l. 

1-3  +  0  -4    +18  +  0    -7    +12 

+  3        +3  +  0  -9-36-27 

-3  -3  -0+9  +  36+27     (x'-^x'=x') 

+  1  +1    +   0-3     -12  -9 


^4  +  0-3^'- 12^-    9;    6ri;'+8:?;  +  3 

The  quotient  is  therefore  x^  ~3x'^ —  12x~9,  and  the 
remainder  Qx"^  -\-Sx  -\~3. 

4.    Divide  x^ -Sx^-5x'+2x'+5x^+4:x''  +  l  by  a;'+2:«;-l. 

The  zero  coefficient  in  the  divisor  may  be  inserted,  or  it 
may  be  omitted  and  allowance  made  for  it  in  the  2^-line. 
See  Exams.  4  and  5  in  Multiplication. 

1-3+0-5+2+5+4+0+1 
-2+6+4-4-6+2 

1-3-2+2+3-1 


-2 
+  1 


1 


-3-2  +  2+3-1;    0  +  5  +  0 

(x^  -^  a^  ^=  a^).     The  quotient  is  therefore  x^  - 
+  2x^  +  ^x  —  1,  and  the  remainder  5rr. 


■?>x'-2a^ 


MULTIPLICATION    AND    DIVISION. 


33 


5.    Divide  10a;'- 11:?;' -  3a;*  + 20r'-|- 10^^' +  2 
Arranging  as  in  the  ordinary  method,  we  have 


10- 

-11-3  +  20 

+  10+    0+    2 

+  3 

6-3-    6 

+  12 

-2 

-4+    2 

+   4-8 

+  2 

+   4 

-2-4+8 

5 

2- 

-    1-2+    4 

24-12+10 

Quotient  =  2o(^  -  x^ —  2x  +  4:- 


24:r'^-12a;  +  10 
5^'-3;r'^  +  2:z;-2' 


We  first  draw  a  vertical  line  with  as  many  vertical 
columns  to  the  right  as  are  less  by  unity  than  the  number 
of  terms  in  the  divisor.  This  will  mark  the  point  at  which 
the  remainder  begins  to  be  formed.     We  then  divide  10  by 

5,  and  thus  obtain  the  first  coefficient  of  the  dividend.  We 
next  multiply  the  remaining  terms  of  the  divisor  by  the 
2  thus  obtained.  Adding  the  second  vertical  column  and 
dividing  by  5,  we  obtain  —1 ;  we  multiply  by  the  —1,  add 
the  next  column,  and  divide  the  sum  by  5,  and  so  on  for 
the  others. 

This  method  is  not,  however,  always  convenient.  If  the 
first  term  of  the  dividend  be  not  divisible  by  the  first  term 
of  the  divisor,  the  work  would  be  embarrassed  with  frac- 
tions.   We  may  then  proceed  as  in  the  following  examples  : 

6.  Divide  ^^-3^*  +  ^'  +  3^'  —  a;  +  3  by  2x^  +  x^—Zx-\-l. 

Let2rj;  =  3/,  or  x^=^^' 
A 

Substitute  ^  for  x  in  the  dividend  and  divisor,  and  we 
A 
have 


34 


MULTIPLICATION    AND    DIVISION. 


25       2*       2'       2'       2  '    2'       2'       2 

_y^-2x3y^+2V+2^x3y^-2\y+2^x3 

2^ 
.  y^+.V^-2x3y+2^ 
2^ 

Dividing  y^-6/+V+2V-16y  +  96  by  y3+y2_6y+4, 
by  the  ordinary  method,  and  the  quotient  by  2^,  we 
have 

y^-7y  +  17       1  ,  39y^-114y-28 
2^  2^  '    y^  +  y^-6y  +  4' 

Substituting  for  y  its  value  2:r,  and  simplifying,  we  get 
^_7^  ,  17_1       39.y^-57rr-7 
2       4  "^  8 


^ 


3       8     2^'  +  n;2-3^  +  l 

By  comparing  the  dividend  of  A  with  the  original  ques- 
tion, we  find  that  we  have  multiplied  the  successive  coefii- 
cients  of  the  dividend  by  2^,  2\  2^  etc.,  and  we  have  mul- 
tiplied the  successive  coefficients  of  the  divisor,  omitting  the 
first  terr}i^  by  the  same  numbers.  Dividing  then  by  Horn- 
er's division,  we  get  the  coefficients  1,  —7,  17;  and  for 
coefficients  of  remainder,  —39,  114,  and  28.  The  first  three 
of  these  divided  by  2,  2^  2^  are  the  coefficients  of  x^,  etc. ; 
and  —39,  etc.,  are  divided  by  1,  2,  2^  Hence,  the  work 
will  stand  as  follows : 


12         4         8 


x+    ^-^2x^  +  x''-?>x+l 
16     32  12       4 


1  -6     +4 

+  24 

-   16  +  96 

-1 

-1     +7 

-17 

+  6 

+  6 

-42 

+  102 

-4 

-   4 

+    28-68 

1-7     +17 

-39 

+ 114  +  28 

1  -6    -f4 


MULTIPLICATION    AND    DIVISION. 


35 


Quotient  >K  ==  £^  - 1^  +  1^  -  1 

^  2       4        8       8 


39  x' 


lUx 


28 
4 


(X 

4  " 


17      1 


2x'  +  x''~3x  +  l 
S9x'-57x~7 


7. 


Divi( 

ie5ar*  +  2by  3.r^- 

5a-^     0        0          0 
1         3        9        27 

2a; +  3. 

0+       2 
81        243 

+  2 
-9 

5        0        0 
10  +  20- 
-45- 

0 

-  60 

-  90 

0+   486 
-280 
+  225  +  1260 

5  +  10-25- 

-140 

-   55  +  1746 

8     2x^  +  x'-Sx+l 


2^3:r^-2:r  +  3 


Coeffs.  of  Quotient  =  --\ - 

3       3'       3''        3' 


55- 


■2    +9 


1746- 


/^        ,•        .  ox      I     ±\J  X  ZiO  X 

Quotient  = —  - 

3  9  27 


3-2  +  3 

55^  —  582 


140       1 


81       81     3^'-2a;  +  3 


Ex.  13. 

Divide : 

1.  6r?;^  +  5.'^'^  -  17r^  -  6^2  + 10:^-2  by  2:^2  + 3;r-l. 

2.  5^'  +  6:x;^+l  by  r^'  +  2^+l. 

3.  a'-6a  +  5by  a'-2a  +  l. 

4.  x^  —  4:x^if  —  d>x''if  —  11  xi/  —  12y^  hj  x'  —  ^xy  —  3y\ 

5.  a^  —  Za^x''  +  3a'x^~x^hY  a^~?>a^x  +  3ax^  —  x\ 

6.  4cx^  +  ?>x^~?>x+lhY  x''  —  2x  +  3. 


*■  It  will,  in  general,  be  more  convenient  to  multiply  the  dividend  by 
such  a  number  as  will  make  its  first  term  exactly  divisible  by  the  first 
term  of  the  divisor,  and  afterwards  divide  the  quotient  by  this  multiplier. 


36  MULTIPLICATION    AND    DIVISION. 

8.  a:^  —  x^y  +  x^if'  —  x^y^  +  ^y^  —  y^  by  x^  —  3/^. 

9 .  Multiply  x"—  ix^a+Qx""  a"—  4 xa^ + a*  by  ^' + 2 :ra + a^ 

and  divide  tbe  product  by  x^  —  2x^a-{-  2xa^  —  a*. 

Divide  : 

10.  x^  —  ax* -{-hx^ —  hx"^ -\- ax —Ihj  x~l. 

11.  6^:^+7:^;'+ 7^' +6^' +  6^ +  5  hy2x'  +  x+l. 

12.  60(a;'  +  y*)  +  91^y(:r2-y')  by  12^^- 13^?/  + 5yl 

13.  6x'  -  481^^  +  Idx*  +  81:^^-  81:z;2  +  86:?;  -  481 

by  ^'  -  80. 

14.  6x^-x'  +  2x'~2x^  +  2x^+19x  +  6  by  Sx^  +  4:X+1. 

15.  a(a  +  2Z>7-^(26^  +  Z^/by  (a-Z>)l 

16.  (a;  +  y)^+3(^  +  y)^2  +  3(.r  +  y)z^  +  2^ 

hy(x  +  yy  +  2(x  +  y)z  +  z\ 

17.  10 :z;^'  +  10:?;"  +  10.^•^  -  200  by  :?;^  +  :?;'-  a;  +  1. 

18.  bmx*  +  (5n  +  cm)  x^  +  C7i:r^  +  ahx  +  a(?  by  ^a;  +  c. 

19.  Multiply  1  +-1^^-  18:r'  by  l-^fx''  +  ^x\  and  divide 

the  product  by  1  +-2^^  —  3^^. 

Find  the  remainders  in  tbe  following  cases : 

20.  (x^  +  ^x'  +  ^x  +  b)-r-{x-2), 

21.  (^'-3^'  +  :z;-3)-f-(:2;-l). 

22.  {x*  +  ^a^  +  ^x  +  S)-^{x  +  2). 

23.  {21x*-y')-^{^x-2y), 

24.  (3a;^  +  5:r*  -  ?>x^  +  7a.-'  -  5a;  +  8)  ^  (o;^  --  2x). 

25.  (5a;*  +  90a;^  + 80a;'-  100a;+  500)  -^  (a;+  17). 


MULTIPLICATION    AND    DIVISION. 


37 


^  8.    The  following  are  examples  of  an  important  use  of 
rner's  Division  : 

1.    Arrange, o;^  —  Qx^  -{-7x~~5  in  powers  of  x  —  2. 


2 

1 

-6 

2 

+  7 
-8 

-5 
—  2 

2 

1 

-4 
2 

-1; 

-4 

-7 

2 

1 

-2; 
+  2 

-5 

1; 

0 

Hence,  x' -Qx'  +  lx-^^--  (x~2y  ~5(x~2)-7; 
or,  as4t  is  generally  expressed, 

x'-6x'+7x-5  =  y'-57j-7if9/=x~2. 
2.    Express  x*  +  12x^  +  4:7x''  +  66x  +  2S  in  powers  of  a;+3. 


1 

12 

47 

66 

28 

-3 

-3 

-27 

-60 

-18 

1 

9 

20 

6; 

10 

-3 

-3 

-18 

-  6 

1 

6 

2  • 

0 

-8 

-3 

-   9 

1 

3; 

-   7 

-3 

-3 

1; 

0 

Hence,  x'  +  12^^-^  +  ^7 x'  +  66:^  +  28  =  y*  - 
if  7/=^x-\-3. 


7y^+10, 


After  a  few  solutions  have  been  written  out  in  full,  as  in 
the  above  examples,  the  writing  may  be  lessened  by  omit- 
ting the  lines  opposite  the  increments  (—2  in  Exam.  1,  and 


38 


MULTIPLICATION    AND    DIVISION. 


3  in  Exam.  2),  the  multiplication  and  addition  being  per- 
formed mentally.  The  last  example,  written  in  this  way, 
would  appear  as  follows  : 

47         66         28  ^ 
20  6       (10) 

2         (0) 


--3 


12 

9 
6 
3 

(0) 


Ex.  14. 

Express  : 

1.  x^  —  bx^  +  3:r  —  8  in  powers  of  ^  —  1. 

2.  ^^  + 3^^ +  6^  + 9  in  powers  of  ^'+ 1. 

3.  x'^  —  8r^  +  24a;'  —  ?>2x  +  97  in  powers  of  ^  -  2. 

4.  r?;*  + 12:?;^  + 5rr' —  7  in  powers  of  a;  +  2. 

5.  3:r^  — :?;^  +  4a;' +  5:r  —  8  in  powers  of :?;  — 2. 

6.  x^  ~  7^:^  +  11^'  —  7:?;+  10  in  powers  of  r^—  If. 

7.  x^  —  2x^  —  4r?;  +  9  in  powers  of  ^  —  -|. 

8.  x^  —  ^x'^y  -{-^ xy^  ~^lf  iii  powers  of  ^  —  2y. 

9.  x^—bx^y-\-b xy^  —  if  in  powers  oi  x  —  y. 

10.  ^x^  -\-Vlx^y  -^  lOrry'  +  83/^  in  powers  of  2:?;  +  y. 

11.  r?;^  — -|r^^  +  f  :r  —  ^  in  powers  of -|-a;  —  ^. 

12.  :?;*  + 80;^  — 15^  — 10  in  powers  of  ^  +  2. 


CHAPTER  II. 

Symmetey. 

;  §  9.  An  expression  is  said  to  be  symmetrical  witt  respect 
to  two  of  its  letters  wlien  these  can  be  interchanged  by 
substituting  each  for  the  other  without  altering  the  expres- 
sion. 

Thus,  a^  -}-  a'^x-{-  ax^  -f  x^  is  symmetrical  with  respect  to 
a  and  x,  for  on  substituting  x  for  a  and  a  for  x  it  becomes 
x^ -{- x"^  a  ^  xa^  ^  a^ ,  which  differs  from  the  original  expres- 
sion merely  in  the  order  of  its  terms  and  of  their  factors. 
So,  also,  x^  +  cc^x  +  <^^  +  ^^^  is  symmetrical  with  respect  to 
a  and  h,  for  on  substituting  h  for  a  and  a  for  h  it  becomes 
x"^  -{'If^  x^ha-{-  a^  x^  which  is  identical  with  the  given  expres- 
sion. On  interchanging  x  and  a,  x^-]-a?x-\~ah'\-b'^x  becomes 
a^ -]- x'^ a-^- xh -{-l)^ a  ]  this  is  not  the  same  as  the  given  ex- 
pression, which  is  therefore  not  symmetrical  with  resj)ect 
to  X  and  «.  In  like  manner,  it  may  be  shown  that  this 
expression  is  not  symmetrical  with  respect  to  x  and  h. 

An  expression  is  symmetrical  with  respect  to  three  or 
more  of  its  letters  if  it  is  symmetrical  with  respect  to  each 
and  every  pair  of  these  that  can  be  selected. 

Thus,  x^-^-'if  +  z^—Zxyz  is  symmetrical  with  respect  to 
X,  y,  and  z,  for  it  remains  the  same  on  interchanging  x  and 
y,  or  y  and  z,  or  z  and  x ;  and  these  are  all  the  pairs  that 
can  be  selected  from  x,  y,  and  z.     So,  also, 


40  SYMMETRY. 


and      {x  -  a){h  -  cj  ■\-  {x-l){c  ~  aj  -\-  {x-  c){a---hy 

are  each  symmetrical  with,  respect  to  a,  Z),  and  c, 

but        (j^h -^-IP' c -\- &  a 

and      (:^-a)(a— Z>)'  +  (^  — ^)(^  — c?)'  +  (r?;  — ^)(c— a)' 

are  not  so. 

oh'\-ac-\-ad-\'hc-^hd^cd 
is  symmetrical  with  respect  to  a,  ^,  ^,  and  d, 
but        a5 +  Z>(?  +  <?c?+c?^ 

is  not  symmetrical  with  respect  to  these  letters,  for  on  inter- 
changing a  and  h  it  becomes 

oh-\'aG -\- cd-\-hd. 
So,  also, 

'bcd-\-  acd-\-  ahd-\-  ahc 

is  symmetrical  with  respect  to  a,  Z>,  <?,  and  d,  as  may  be  seen 
at  once  by  writing  it  in  the  form 


\a      0      c      aj 


An  expression  is  cyclo-symmetric  with  respect  to  three 
of  its  letters,  a,  S,  and  (?,  if  it  remains  the  same  expression 
when  a  is  changed  into  Z>,  h  into  c^  and  c  into  a. 

Thus,  a^h-^y^c-^-c^a  is  cyclo-symmetric  with  respect  to 
the  cycle  (ahc),  for  on  changing  a  into  b,  h  into  (?,  and  c  into 
a,  it  becomes  Z>^c?  +  6'^a-f-a^&,  which  differs  from  the  orig- 
inal expression  only  in  the  order  of  its  terms. 

(a  —  Ijf  +  (^  —  ef  -\-{g  —  of  is  not  symmetrical  with  re- 
spect to  a,  5,  and  c,  for  interchanging  a  and  h  changes  it 
into  (h-ay+{a- cj +{c- bf  which  =-(a~bf-(b-cf 
—  ((?  —  d)^,  and  this  differs  from  the  original  expression  in 


SYMMETRY.  41 


the  signs  of  all  its  terms.  But  this  expression  is  cyclo- 
symmetric  with  respect  to  (ah  c). 

So,  also,  {x-a){a-hy  +  {x--h){b-cy  +  {x-c){c~ay 
is  cyclo-symmetric  with  respect  to  (a  h  c)  but  is  not  com- 
pletely symmetric  with  respect  to  a,  b,  and  c. 

Generally/,  an  expression  is  cyclo-symmetric  with  respect 

to  any  set  of  letters,  a,  b,  c,  ,  h,  Ic,  called  the  cycle 

{ab  c hlc),  if  it  remains  the  same  expression  when  a  is 

changed  into  5,  b  into  c,  ,  h  into  ^,  and  Ic  into  a. 

Thus,  ab-\-bc-\-cd'\-  da  and  ac  +  bd  are  each  cyclo-sym- 
metric with  respect  to  the  cycle  (abed),  but  are  not  com- 
pletely symmetric  with  respect  to  a,  b,  c,  and  d. 

Every  expression  which  is  completely  symmetric  with 
respect  to  a  set  of  letters  is  necessarily  cyclo-symmetric 
with  respect  to  them ;  but,  as  is  seen  by  the  above  exam- 
ples, an  expression  may  be  cyclo-symmetric  without  being 
completely  symmetric. 

Principle  of  Symmetry.  An  expression  ivhich  in  any 
one  form  is  completely  symmetric^  or  is  cyclo-symmetric,  with 
respect  to  any  set  of  letters  will  in  every  other  form  be  com- 
pletely  symmetric,  or  be  cyclo-symmetric,  as  the  case  may  be, 
with  respect  to  these  letters. 

Thus,  a^  +  ¥  +  c^  —  ?>abc  is  symmetrical  with  respect  to 
a,  b,  and  c ;  hence,  it  will  be  symmetrical  when  written  in 
any  other  form,  as,  for  example,  in  the  form 

Again,  (a  —  by  +  (b  —  cf  +  (c~  a)^  is  cyclo-symmetric, 
but  not  completely  symmetrical,  with  respect  to  {ab  c)]  it 
will  therefore  remain  thus  cyclo-symmetric,  but  not  com- 
pletely symmetrical,  under  every  change  of  form  which  may 
be  given  it ;  for  example,  when  it  is  reduced  to 

?.(a-b){b-c){G~a). 


42  SYMMETRY. 


A  symmetric  function   of  several  letters  is  frequently 
represented  by  writing  each,  tyjpe-ternfi  once,  preceded  by 

the  letter  2  ;  thus,  for  a  +  Z)  +  (?+ +  Z  we  write  2  a,  and 

for  ah  -\- ac -\- ad -{• -\-hc-^hd-\- (that  is,  the  sum  of 

the  products  of  every  pair  of  the  letters  considered)  we 
write  SaZ>. 

Ex.  15. 

Write  the  following  in  fall : 

1.  %a^h',    %{a-hy',    %a(h-c)',    :^ab(x-c);    ^a'Pc, 

%{a  +  h)  {c~  a)  {c-h)-    2[(a  +  e)^-Z>^];    and 
Sa(Z)  +  cf^  each  for  a,  h,  c. 

2.  %ahc)  %a^h]  ^d^hc]  %{a  —  h)]  and  5 a^ (a  — ^),  each 

with  respect  to  a,  ^,  c,  d. 

Show  that  the  following  are  symmetrical : 

3.  (x  +  «)(«+  ^)  (^  +  x)~{~ahxj  with  respect  to  a  and  h. 

4.  {a  ~{~  hy  -\~  {a  —  by  with  respect  to  a  and  b,  and  also 

with  respect  to  a  and  —b. 

5.  (ab  —  xyy  —  (a-^-b  —  x  —  y)  [ab  (x-\~y)  —  xy  (a  +  by] 

with  respect  to  a  and  b,  and  also  with  respect  to  x 
and  y. 

6.  a^(b  —  c)^+5^(^— a7+c^(a— ^)^  with  respect  to  a,  Z>,  <?. 

7.  (ac  +  Z)(i)^  +  {be  —  ady  with  respect  to  a^  and  Z>^  and 

also  with  respect  to  c^  and  d'^, 

8.  .T®  +  y®  +  3  :ry  (:r^  +  xy  +  y^)  with  respect  to  x  and  y. 

9.  [.^'^-/  +  3^y(2:r  +  3/)]^+[2/'-^  +  3:.y(2y  +  rt•)]^ 

with  respect  to  x  and  y. 

10.    a(a  +  2^/  +  ^  (Z>  +  2a)^  with  respect  to  a  and  &,  and 
also  with  respect  to  a  and  —  Z>. 


SYMMETRY.  43 


> 


11.  ah  \[{a  +  c){b  +  c)+2c{a  +  h)y  ~  {a-  c)\h  -  cf\ 

with  respect  to  a,  h,  c. 

12.  o?  h^  ^  y^ c}  -{-  c^ a^  -\- 2 ahc  {a  +  Z>  +  e)  with  respect  to  ah, 

he,  ca. 

With  respect  to   what  letters   are    the    following    sym- 
metrical ? 

13.  xyz  +  bxy  +  2{x'  +  7f). 

14.  2{d'x^  +  b''y'')-2ah{xy  +  hy  +  ax). 

15.  {f  -  hy  +  4^^  (/+  hy  +  {2fh  -  2gJ. 

16.  {x  +  y){x-z){y~z)-xAjz. 

17.  a^h''  +  h''  c"  +  c'a^  -2abc{a  +  h  -  c). 

18.  x^  -  y«  +  0«  -  3  {x'  -  /)  (y^  -  2^)  {z"  +  :i-^). 

19.  (a  +  &)2  +  (a  +  c)'  +  (^-^7. 

20.  {a  +  Z^)*  +  {a  -  c?)*  +  (/;  +  c)'  +  («  +  c)\ 

21.  (a  +  ^)*  +  (a  -  cy  +{b  +  cy  +  (a  +  of  +  (c  ~  by. 

Select  the  type-terms  in  : 

22.  a^  +  2ab  +  b'  +  2bc  +  c^  +  2ca. 

23.  a  (b'~  c')  +  b  (c'-  a')  +  c (u'-  P)  +  (a  +  b)  (b+c)  (c+a). 

24.  a (b  +  cy  +  b{c  +  ay  +  c{a  +  by-  12abc. 

Write  down  the  tyjpe-terms  in : 

25.  {x-\-yy',  {x-yy-,  {x  +  yy-x'^  —  f. 

26.  {x  +  yy  +  ^x-yy-,  {x  +  yy  -  Qc  ~yy. 

27.  {x  +  y  +  zy-,  {x-y~zy. 

28.  (a  +  Z>  +  c  +  c/y;  (a^  +  Z>'' +  c' +  c^yi 

29.  {aJ^by-\-ib-\-cy^(c-\-ay, 


44  SYMMETRY. 


§  10.  In  reducing  an  algebraic  expression  from  one  form 
to  another,  advantage  may  be  taken  of  the  principle  of  sym- 
metry ;  for,  it  will  be  necessary  to  calculate  only  the  tyjpe- 
terms,  and  the  others  may  be  written  down  from  these. 

Examples. 

1.  Find  the  expansion  oi{a  +  h-\~C'\-d-\-e-{- y. 

This  expression  is  symmetrical  with  respect  to  a,  ^,  c, 

;  hence  the  expansion  also  must  be  symmetrical, 

and,  as  it  is  a  product  of  two  factors,  it  can  contain 

only  the  squares  o?,  5^,  c^,  ,  and  the  products  in 

pairs,  ah,  ac,  ad,  ,  he,  hd,  ;   so  that  c^  and  ah 

are  type-terms. 

Now  (a  +  5)^  =  a^  +  2  a&  +  5^ ;  and  the  addition  of  terms 

involving  c,d,e, ,  will  not  alter  the  terms  a^-\-2ah, 

but  will  merely  give  additional  terms  of  the  same 
type.     Hence,  from  symmetry  we  obtain 

{a+h-\-c+d+e+ y-=  a^  +  2ah  +  2ac+2ad+2ae+ 

+  5'     +2hc+2hd+2he  + 

+  c^     +2cd  +  2ce  + 

+  c?'    +2de+ 

+  e'     + 

This  may  be  compactly  written 

{%ay  =  %a'  +  2%ah. 

2.  Expand  (a +  5)1 

(i.)  The  expression  is  of  th^ee  dimensions,  and  is  sym- 
metrical with  respect  to  a  and  h. 

(ii.)  The  type-terms  are  a^  a^h. 

Hence,  {a  +  h)^  =  a^  +  h^  -\-  n{a^h  +  Pa),  where  n  is 
numerical. 

To  find  the  value  of  n,  put  a  =  h=^l,  and  we  have 


SYMMETRY.  45 


3.  Expand  (^  +  y  +  zf. 

This  is  of  three  dimensions,  and  is   symmetrical  with 
respect  to  x,  y,  z.     We  have 

{x  +  y  +  zy^[(:x  +  y)  +  zY^{x  +  yy+ 

=  x^ -\- '^  x^  y -\r 

which  are  type-terms,  the  only  other  possible  type- 
term  being  xyz. 
Now,  since  the  expression  contains  2>x^y,  it  must  also 
contain  ^x'^z\    that  is,  it  must  contain   2tx^{y-\-z). 
Hence, 

{x  +  y  +  zy=     x^  +  ^  x^  {y  -f  z) 
.    +y'  +  ^f{z  +  x) 
+  z'+?>z'{x+y) 
+  n{xtjz), 
where  n  is  numerical,  and  may  be  found  by  putting 
x^=^y=zz  =  l  in  the  last  result,  giving 

(1  +  1  + 1)^=1  + 1  +  1 -f  3(1 +  1)  +  3(1  +  1) 

+  3(l  +  l)  +  72. 

Hence,  ?^  ==  6. 

4.  Similarly,  we  may  show  that 

{a  +  h  +  c  +  df^  a^-\-?>a^{h  +  c  +  d)  +  Ucd 
+  Z)'  +  3^^  \c  +  cZ+  a)  +  6c(7a 
+  ^•'^  +  3^^  \d^a-\-l)-\-^dab 
+  d}-\-  3  (i^  (a  +  5  +  c)  +  6  ahc. 

5.  Expand  (a +  Z)  +  ^+ J. 

The  type-terms  are  a^,  a^S,  ahc. 

Expanding  (a  +  Z>  +  (?/,  we  get  o}-\-Zc^h-\-^abc-\' 

Hence,  by  symmetry,  we  have 
(:Sa)^==:Sa^  +  3^a2^  +  G^«Z)^. 


46  SYMMETRY. 


6.  Simplify  {a  +  h-  2cy  +  (b  +  c  ~2ay  +  (c  +  a-  2b)\ 

This  expression  is  symmetrical,  involving  terms  of  the 
types  o?  and  ah.  Now,  a^  occurs  with  1  as  a  coeffi- 
cient in  the  first  square,  with  4  as  a  coefficient  in  the . 
second  square,  and  with  1  as  a  coefficient  in  the  third 
square,  and  hence  6  a^  is  one  type-term  of  the  result ; 
ah  occurs  with  2  as  a  coefficient  in  the  first  square, 
with  —4  as  a  coefficient  in  the  second  square,  and 
with  —4  as  a  coefficient  in  the  third  square,  and  hence 

—  6  ah  is  the  second  type-term  in  the  result.     Hence, 
the  total  result  is  d  (a'^  -\-  h"^  -\-  c^  —  ah  —  hc~  ca). 

7.  Simplify  (x~\-y  +  zf -{- (x  —  y —  zf-^- (y  ~  z  —  xf 

+  {z-x-  y)\ 

This  is  symmetrical  with  respect  to  x,  y,  z\  and  the 

type-terms  are  a^,  3^^y,  ^xyz: 
(i.)  a?  occurs  in  each  of  the  first  two  cubes,  and  ~0(^  in 

each  of  the  second  two  cubes ;  therefore,  there  are  no 

terms  of  the  type  oi?  in  the  result. 
(ii.)  Zoc^y  occurs  in  the ^rs^  and  third  cubes,  and  —^x'^y 

in  the  second  and  fourth ;  therefore,  there  are  no  terms 

of  this  type  in  the  result, 
(iii.)  ^xyz  occurs  in  each  of  the  four  cubes;  therefore, 

2^xyz  is  the  total  result. 

8.  Prove  {a'+h'^+c'+d'')  {w^+x'+y''+z^)-{aw+hx+cy+dzy 

—  {ax — hwy  +  {ay  —  cvif  +  {az — dw^  +  (Ijy  —  cxy 
+  {hz-dxy  +  {cz-dy)\ 

The  left-hand  member  (considered  as  given)  is  symmet- 
rical with  respect  to  the  pairs  of  letters,  a  and  w,  h 
and  X,  c  and  3/,  d  and  z ;  that  is,  any  two  pairs  may 
be  interchanged  without  affecting  the  expression.  As 
the  expression  is  only  of  the  second  degree  in  these 


SYMMETRY.  47 


pairs,  no  term  can  involve  three  pairs  as  factors ; 
hence,  the  type-terms  may  be  obtained  by  consider- 
ing all  the  terms  involving  a,  h,w,x\  these  are  aW, 
d^x^,  l/w^,  y^x^,  —a^vf,  —W-x^,  —2ahwx,  and  are  the 
terms  of  (ax—hvif,  which  is  consequently  a  type- 
term.  From  (ax  —  hivy  we  derive  the  five  other  terms 
of  the  second  member  by  merely  changing  the  letters. 

9.    Prove  that  {x^  —  yzf  -f  (y^  —  zxf  +  (2;^  —  xy^ 

—  Z{p(?  —  yz){y^  —  zx) {z^  —  xy)  is  a  complete  square. 

The   expression   will   remain   symmetrical   if  {x^  —  yz) 
{y'^  -—  zx)(z^  —  xy),    instead   of  being   multiplied   by 

—  3,  be  subtracted  from  each  of  the  preceding  terms, 
thus  giving 

(^'  -  yz)  [{o(?  -  yzj  -  (y'  -  zx)  {f  -  xy)^ 

+  (3/'  -  ^^)  w  -  ^^y  —  (^'  -  ^y)  (^'  -  y^)] 

+  (z'  -  xy)  [(2;'  —  xyj  -  {x^  —  yz)  {if  —  zx)'\ 

=  {x?-  —  yz)x(x^  +  f  +  ^^  —  ^xyz)  -f 

=  {x^  +  f  +  z^  —  3xyz)  (x^  +  y^  +  z^  —  3xyz). 

Ex.  16. 

Simplify  the  following : 

1 .  (a  +  h  +  cf  +  (a  +  b  -  cy  +  (h  +  c-  ay  +  (c  +  a  ~  by. 

2.  (a-b-cy  +  (b-a-cy  +  (c-a-by. 

3 .  (a  +  b  +  c~dy  +  (b  +  c  +  d-ay+(c  +  d+a-by 

+  (d+a  +  b-cy. 

4.  (a  +  5  +  cy  —  a(b-}-c  —  d)  —  b(a-\-c~b)  —  c(a  +  b  —  c). 

5.  (x~\-y-^z-{-ny-{-(x  —  y  —  z-\-ny-\-(x-—y-\-z  —  ny 

+  (x  +  y  —  z  —  ny. 

6.  (a  +  b  +  cy+(a  +  b-cy  +  (b  +  c-~ay+(c  +  a-by\ 

7.  (x  -  23/  -  3^)'^  +  {y-2z  -  3xy  +  (z  -  2x  ~  3y)\ 


48  SYMMETRY. 


I 


8.  (ma  -\-  nb  -{-  rcf  —  (ma  -\- nh  —  rcf  —  (nh  +  re  —  mdf 

—  (re  +  ma  —  nhy. 

9.  a(h  +  c)(b'  +  c'-  a:')  +  b(c  +  a)  (c'  +  a' ~  b') 

+  c(a  +  b)(a'  +  b'-c'), 

10.  (ab  +  bc  +  caf  —  2abc(a  +  b  +  c). 

Prove  the  following : 

11.  (ax  -\-  by  -\-  czf  +  (bx  -j-  cy  -{-  azf  +  (ex  +  ay  +  bzj- 

+  (ax  ■\-ey  -\-  bzf  +  (ex  A^by  -\-  azf  +  (bx-{-ay  +  ezf 
=  2(d'  +  b''  +  e^)(x^+y^  +  z^) 
+  4  (ab  -{-be  -\-  ed)  (xy  +  yz-\-  zx). 

12.  (a  +  b  +  ey  +  (b  +  c-ay+(e  +  a  —  by  +  (a  +  b  —  cy 

=^4:(a'  +  b'  +  e')  +  24  (a'b'  +  b'e'  +  c^a^). 

13.  (a  +  ^  +  (?)^  =  2a*  +  42a^^>  +  6Sa^^^  +  122a^^^. 

14.  (:^ay  =  :$a'  +  4c:^a'b  +  6^a'b'  +  12^a''bc+24:^abed. 

15.  (a^  +  ^=^  +  c^)^  +  2(ab  +  be  +  eaf 

-S(a'  +  b'  +  e')  (ab  +  be  +  eaf 
=^(a'  +  b'  +  c'-Sabey. 

16.  (a  -  by(b  -  cf  +  (b-  cy  (e  -  ay  +  (c  -  ay  (a  -  by 

=  (a'  +  b^  +  c^-ab-ae-  bey, 

17.  (2a-b-cy(2b-c-ay  +  (2b-e-ay(2c-a-by 

+  (2e-a-by(2a-b-ey 

=  ^(a^  +  b''  +  c^-  ab-bc-  eay, 

18.  (ar^  -\-2brs  -\-  es^) (ax^  +  2 bxy  +  cy'^) 

—  [arx  +  b  (ry  +  sx)  +  csy'\  ^  =  (ae — b'^)  (ry  —  sxy. 

19.  (a^  +  ab  +  b')(e^  +  ed+d') 

=  (ac-{-ad-{-bdy-}-(ae-}-ad-]-bd)  (be~ad)-{-(be—ady. 

20.  Show  that  there  are  two  ways  in  which  the  given 

product  in  the  last  example  can  be  expressed  in  the 
form  p^  +pq  +  2'^  and  two  ways  in  which  it  can  be 
expressed  in  the  form  p^  ~  p(]_  +  (f- 


I 


THEORY    OF    DIVISORS.  49 

21.  6(w^  +  ^'  +  y' + z'f  ==(w  +  xy  +  (w-  xy + (iv + 7/y 

+  (w-  yy  +  {w  +  zy  +  (iv  -  zy  +  {x  +  yy+  {x~yy 

+  {x+zy+{x-zy+{y+zy+{y-zy. 

22.  \[{a  +  h  +  cy  +  {a-h-cy  +  (h-c-ay+{c-a-hy] 
=^^[{a-\-h+(}f+{a-h-cy+{h^c~ay+{c-a-hy] 
X^[{a+b  +  cy+  {a-h-~  cy+  (b  -c-ay+  {c-a-hy\ 

Theoky  of  Divisors. 

Any  expression  which  can  be  reduced  to  the  form 

ax''  +  bx''-'^  +  ex''-''  + +  hx+'k, 

in  which  ?z  is  a  positive  integer,  and  a,  b,  c,  ,  h,  Ic,  are 

independent  of  x,  is  called  a  Polynome  in  x  of  degree  n. 

The  expressions /(rr)**,  Fixy^  <^  (a;)"*,  are  used  as  general 
symbols  for  polynomes ;  the  exponents  n  and  m  indicate 
the  degree  of  the  polynome. 

Theorem  I.  If  the  polynome  /(xY  be  divided  by 
X  —  a,  the  remainder  will  be /(a)**. 

Cor.  1.  f(pcy—f{aY  is  always  exactly  divisible  by  x—a. 

Cor.  2.  If  f{ay  =  0,  f(xy  is  exactly  divisible  hj  x  —  a; 
that  is,/(:r)"'  is  an  algebraic  multiple  of  ^  —  a. 

Cor.  3.  If  the  polynome  f(xy,  on  division  by  the  poly- 
nome (fi  (xy,  leave  a  remainder  independent  of  x,  such 
remainder  will  be  the  value  off(xy  when  <^  (xy  —  0. 

Examples.     Theorem  I. 

1.    Find  the  remainder  when  x^  —  7x^-\-lSx^—16x''-\-9x—12 
is  divided  by  :?;  —  5. 
The  remainder  will  be  the  value  of  the  given  polynome 
when  5  is  substituted  for  x.     (See  §  3.) 


50  THEORY    OF    DIVISORS. 


1 

-7    +13 
5-10 

-16 
15 

+  9 
-5 

-12 

20 

1 

-2          3 

-1 

4; 

8 

Hence,  the  remainder  is  8. 

2.  Find  the  remainder  when  (x  —  a)^  -\~  (x  ~  lif  +  (a  +  hy 

is  divided  by  x-\-  a. 
For  X  substitute  —a,  then 

(-  2af  +  {-a-hy+  {a  +  hy  =  -^a\ 

3.  Find  the  remainder  when 

C(^  +  o^  +h^  +  (x  +  a){x  +  h)  {a  +  h) 

is  divided  hj  x-\-  a-{-h. 
For  X  substitute  —  (a  +  J),  and  we  obtain 

-{a  +  hj  +  a^  +  b^  +  ah{a+h)=^-2ah{a  +  h). 
See  Formula  [6]. 

4.  Find  the  remainder  when 

(x"  +  2ax  -  2ay  (x"  -  2ax  -  2a')  +  32  (x  -  a)'  {x  +  a)' 

is  divided  hj  oc^  —  2al 
x^  —  2o?  may  be  struck  out  wherever  it  appears. 
This  reduces  the  dividend  to 

(2  axf  (-  2  ax)  +  32  (^  -  a)'  (x  +  a)* 

=  -16a'x'  +  32(x'-a'y. 
In  this  substitute  2  a'  for  rr',  and  it  becomes 

-64a«  +  32a^  =  -32a^ 

which  is  the  required  remainder. 

Ex.  17. 

1.  Find  the  remainder  when  dx'+60x^+  54: x""  —  60 x  +  58 

is  divided  by  a;+  19. 

2 .  Find  the  remainder  when  pa^  —  Sqx"^ + 3  r:r — s  is  divided 

hj  x  —  a. 


THEORY    OF    DIVISORS.  51 

3.  What  number  added  to 

4^^+  34:^^  +  58:^;^  +  21^^  -  123:r  -  41 

will  give  a  sum  exactly  divisible  by  2^+  13. 

4.  What  number  taken  from 

10 :r^^  -  20a;«-  10:?;«  -  0.89 ^r*  -  8.9^'  +  20 

will  leave  a  remainder  exactly  divisible  by  10^^—11  ? 

Find  the  remainders  from  the  following  divisions  : 

5.  (:^+l)'-:?;'-f-:^'  +  l;   (x  +  a  +  Sy-(x-i-a  +  iy-^x+2. 

7.  (x+lf  +  x'  +  (x-iy-^x-2.. 

8.  (x-  of  {x  +  aj  +  (^^  -  2 h'y  -^x^  +  b\ 

9 .  (x^  -\-ax-\'  a^)  {oc^  ~-ax~\-  a^) 

-  {x''  -?>ax  +  2a') {x''  +  ^ax  +  2a')~r-  x"  +  2al 

10.  (9a'+6a&  +  4Z^^)(9a'-6a5  +  4Z>')(81a*-36a'^'  +  16Z)^) 

-^  (3a -25)1 

11.  d^-ix  —  af -\-lP' {x—'hf  ^x  —  a  —  h, 

1 2 .  {ax  +  hyy  +  a^  if  +  ^^  ^^  —  3  abxy  (ax  +  by) 

-^{a  +  b){x  +  y). 

13.  x^  +  a''^  +  5^  —  3  a5:z;  -^  :r  —  a  +  ^  ;    also,  -^  x-\-  a  —  h] 

also,  ^  X  —  a  —  h. 

14.  Any  polynome  divided  by  :r— 1  gives  for  remainder 

the  sum  of  the  coefficients  of  the  terms. 

Examples.     Cor.  1. 

1.    a;^  +  y^  is  exactly  divisible  hj  x-^ry- 

In  ''x^—a^  is  exactly  divisible  hj  x—a,''  substitute  —y 
for  a. 


52  THEORY    OF    DIVISORS. 

2.  mx^  —px^ -\- qx-\-r}i-{-p -{- q  is  exactly  divisible  hj  x-{-l. 
This  may  be  written 

{mx^  —px^  +  qx^  —  [m  (—  1)^  —p(^—Yf-^q (—  1)] 
is  exactly  divisible  by  ^  —  (—  1). 

3.  {a?-\-^xy-\-  4/)'  +  (:r'  +  2:^3/  +  4?/')^  is  exactly  divis- 

ible by  (:?;  +  2y)l 
For  (x^  +  G^ry  +  4y^)^  —  {;—x^  —  1xy  —  4^/^)^  is  exactly 
divisible  by  {x^ -^  ^  xy  -\- ^  y^)  —  (—  ^'^  —  2  ^y  —  4  y^) , 
which  is  2  (a;'  +  4a;y  +  4y')  =  2 (a;  +  2y)l 

Ex.  18. 

Prove  that  the  following  are  cases  of  exact  division  : 

1.  :r'"+' +  y'"+^ -^  ^  +  y ;  x"-""  —  y'''' ^  x -\- y . 

2.  :^;^^  +  y^2-^:^;*  +  y*;  ^'«  +  y'*' -^  a;' +  y^  also, -^  ^^'^  +  y'' ; 

also,  -^x^-\-  yl 

3.  (a:r  +  %)^  +  (^o;  +  ay7 -^  (a  +  5)  (:^  +  y). 

4.  {ax  +  %  +  C2;)^  —  (^:?;  +  <?y  +  <^2;)"^ 

-^  (a  —  ^)  5;  +  (^  —  (?)  y  +  (^  —  a)  2;. 

5.  (2y-a;)"  — (2:?;  — y)^-^3(y  — a;). 

6.  (2y-:ry"+^+(2^-y)'"+'-^y  +  :z;. 

7 .  {my  —  nxf  —  {mx  —  nyY  -^  (m  -\-  n)  (y  —  x). 

8.  {x  +  yy  +  (x~yy-^2{x'  +  f). 

9.  (x'  +  xy  +  yy  +  (x'-xy  +  yj^2{x'  +  y'). 

10.  (a  +  &7-(a-Z>y^2&(3a'  +  ^>'0. 

11.  (x'  +  5Z>:z;  +  ^>')^  +  (x' -hx  +  bj -^2(x  +  h)\ 

12.  (a  +  ^y"+'  +  («-^r+'-2(a^  +  ^0- 

13.  [:z;^  +  3:ry(rr-y)  -y^]H  [:6^^-  9^y(a,-  -  y)  -  y']' 

^2(^-y)l 


THEORY    OF    DIVISOKS. 


14.  ^x'-5x'  +  4:x-2-^x—l. 

15.  Any  polynome  in  x  is  divisible  by  ^—1  when  tbe 

sum  of  the  coefficients  of  the  terms  is  zero. 

16.  Any  polynome   in  x  is  divisible  hj  x-j-1  when  the 

sum  of  the  coefficients  of  the  even  powers  of  x  is 
equal  to  the  sum  of  the  coefficients  of  the  odd  pow- 
ers.    (The  constant   term   is   included    among   the 
'  coefficients  of  the  even  powers.) 

Examples.     Cor.  2. 

1.  Show  that  a(a-j-2by  —  b(2a-{-hy  is  exactly  divisible 

by  a-{-b. 

By  Cor.  2,  the  substitution  of —h  for  a  must  cause  the 
polynome  to  vanish. 

Substituting,  a(a  —  2ay  +  a(2a  —  af  =  —  a^  +  a^  =  0. 

2.  Show  that 

(ah  —  XT/Y  —  (a-{-b  —  x  —  T/)  [ab  (x  +  y)  —xi/  (a  +  b)] 
is  exactly  divisible  by  (x  ~  a){y  —  a),  also,  by 
(x-b){y-b). 

For  X  substitute  a,  and  the  expression  becomes 

{ab  -  ayj  -ib-y)  {ah  (^  +  3/)  -  ctyifL  +  by\ 
^a\b~yy-{b-y)\a\b-y)-\^^. 

The  expression  is,  therefore,  exactly  divisible  \)j  x  —  a. 
But  it  is  symmetrical  with  respect  to  x  and  y,  hence 
it  is  divisible  by  y  —  a\  and,  as  x  —  a  and  y  —  a  are 
independent  factors,  the  expression  is  exactly  divisi- 
ble by  {x  —-  d){y  —  a).  Again,  the  given  expression 
is  symmetrical  with  respect  to  a  and  b ;  hence,  mak- 
ing the  interchange  of  a  and  b,  the  expression  is 
seen  to  be  divisible  by  (x~b){y  —  b). 


54  THEORY    OF    DIVISORS. 


3.  Show  that  6  (a'  +  b'  +  c')-^5  (a'  +  b'  +  c')  (a'  +  b'  +  c') 

is  exactly  divisible  by  a-\-b-{-c. 

For  a  substitute  —(b-{-c),  and  the  result,  which  would 
be  the  remainder  were   the   division   actually  per- 
formed, must  vanish. 
6[~(b  +  cy+b'  +  c'] 

-~6[-(b  +  cf  +  ¥  +  c']  [(b  +  cf  +  b'  +  c'] 
=-.  6[-(b  +  cf  +  b'  +  c']  +  SObc(b+  c)(b'  +  bc  +  c'). 

See  [1]  and  [6]. 

The  expansion  being  of  the  fifth  degree,  and  symmetri- 
cal in  b'  and  c,  it  will  be  sufficient  to  show  that  the 
coefficients  of  &^  ¥c,  h^  (?  vanish,  the  coefficients  of 
b'^c',  bc^,  &  being  the  coefficients  of  the  former  terms 
in  reverse  order.  Calculating  the  coefficients  of  these 
type-terms,  we  get 

^\-hb'c-VdW&- ]  +  30(&*^  +  2^>''c'+ ), 

which  evidently  vanishes.     Hence,  the  truth  of  the 
proposition. 

4.  If  a-t-^^-fc-0, 

\{a'-\-b'^c')^\{a'-^y'  +  c')^\{a'-Vb'  +  c% 
In  the  last  example  it  has  been  proved  that  the  differ- 
ence of  the  quantities,  here  declared  to  be  equal,  is  a 
multiple  oi  a-\-b-\-c^  that  is,  in  this  case,  a  multiple 
of  zero.  Hence,  under  the  given  condition  they  are 
equal. 

Ex.  19. 

Prove  that  the  following  are  cases  of  exact  division  : 

1 .  {ax  —  byj  +  ibx  —  ayf  —  {c^  -f  b^)  {x"  —  i/)-^a,b,  x,  y, 

a  +  5,  x—y. 

2.  ax^  —  (a^  -^  b) x'^  -\- b'^ -^  ax  —  h.     (Substitute  ax  for  b.) 


THEORY    OF    DIVISORS.  55 

^{{ax+bijy~{a  —  h){x  +  z){ax+by)  +  {a~hyxz^x+y. 
\  (ax — hyy—  {a  +  h){x-\-z)  (ax  —  by)  +  (a-^by xz-^x-\-y. 

4.  Qa^x^  —  4iax^  —  10axy — ?>  a^ xy -\-2 x"^ y -{-by'^ -^  2ax — y. 

5.  l.2a'x~  5.494a' ^'  +  4.8a' ^'  +  ^.^ax'  ~  x"" 

6.  x^  ^  x^y'^  ^x^y -^1^  ^x^ -\-y, 

7.  (c  -  d)a^  -\-^(bG  -bd)a  -\-  "^(1?  c  -b^  d)  -^  a  -\-Zb, 

8.  x(x--^yf-\-y(-^x-yy-^x-y. 

9.  a(a  +  2^)'-Z>(^  +  2a7-f-a-5,  also-^a  +  Z>. 

•  10.    a^— 2a'5  +  a'^)'  +  ^'^'-2a&^''+Z>':r'-f-(a-^)(^  +  a). 

11.    a(b  -  cj  ^b(c  ~  aj  +  c(a-by'  -^  (a-b),  (b-6), 
(c-a). 

\  12.    d(b-c)  +  b\c-a)-^c\a-b)-^(a-b),  (b  -  c), 
{c  -  a). 

13.    a'  {b-c)  +  b'  (c  —  a)  +  c\a  -  b) -^  (a  -b),  (b  -  c), 
(c  -  a). 

14.  (a-  by  (c  -  dy  +  (b-  cy  (d  -  ay  -(d~  by  (a  -  cy 

-i-  (a  —  b),  (b  —  c),  (c  —  d),  (d  —  a). 

15.  [(a-by  +  (b-cy  +  (c-  ay][(a-byc'+  (b-  cya' 
+  (c-  ayb']  -  [(a  -byc  +  (b-  cya  +  (c-  ay  by 
-f-  (a  —  b),  (b  —  c),  (c  —  a). 

16.  (x  +  y){y  +  z)(z  +  x)  +  xyz-^x-\-y  +  z. 
IT.    abid"  -  b'')  +  bc(b''  —  c')  +  ca(c'' -  a') -^  a+b  +  c. 

18.  (ab  —  be  —  cay  —  d^b'^  ~  b'^ c^  —  c^ a^  ^  a+  b  —  c. 

19.  (a  +  2by  +  (2b  -Scy-(Sc~  ay  +  a'  +  Sb'  -  27c' 
-r-a  +  2b  —  Bc. 

20.  a'b'  +  b'c'  +  c'a'  -Sa'b'c''  ^  ab  +  be  +  ca. 


56 


THEORY    OF    DIVISORS. 


Examples.     Cors.  3  and  2. 
Find   the    value    of  4:X^  +  ^x^  —  bx"  +  2?>x  +  ^  when 

Since  2^^  —  3^  +  4  =  0,  we  have  simply  to  find  the 
remainder  on  division  by  2x^  —  3:z;  +  4;  and,  if  it  is 
independent  of  x,  it  is  the  value  sought,  Cor.  3. 


4    0 

9 

-    5 

23 

6 

3 

6 

9 

15 

-  3 

-4 

-8 

-12 

-20 

4 

2 

2     3 

5 

-  1; 

0 

10 

Hence,  the  required  value  is  10. 

2.    What  value  of  c  will  make  x^  —  bx^-{-lx  —  c  exactly 
divisible  by  x—2. 
If  2  be  substituted  for  x,  the  remainder  must  vanish, 
Cor,  2. 


1 

-5 
2 

7    -c 
-6        2 

1 

-3 

1-2-c 

bx^  +  cx^—2{)x''  +  l^x 
1. 


Hence,  2  —  <?  =  0,  or  c  =  2. 

3.  What  value  of  c  will  make  6^^ 
—  5  vanish  when  2x^  =^?>x 
By  Cor.  3  the  remainder  must  vanish  when  the  given 
polynome  is  divided  by  2x'^  —  ?>x-\-l.  We  may 
divide  at  once  and  find,  if  possible,  a  value  of  c  that 
will  make  both  terms  of  the  remainder  vanish ;  or, 
we  may  first  express  cx^  in  lower  terms  in  x^  and 
then  divide  and  find  the  required  value  of  c  from 
the  remainder. 


THEORY    OF    DIVISORS. 


57 


-    First  Method  (see  page  31). 

6-10         Ac       -160 

304 

-160 

3 

18        24     12e  +  36 

36^-420 

__  2 

-12            -16 

-8^-24 

-24^  +  280 

6       8,  4c  +  12,  12^-140; 

28^-140 

-24^+120 

Hence,  28^=  140  and  24  c  =  120.    Both  of  tliese  are  sat- 


isfied  by  c  =  5. 

Second  Method.     x'  =  ^x(?,x- 

-  1)  =  f  a,-^  --  ^x 

=  f(3a;-l)-|s;  =  2|.a;- 

-|_|.X=13^^ 

-f 

.•.  C3?  =  l^cx  —  fe. 

Substituting  for  cx^  in  the  given  polynome,  it  becomes 

6x'—5x'  —  20:^;'^  +  (If  c  +  19)x  -3^-5. 
Divide  and  apply  Cor.  3. 


6 

-10 

0 

-160 

28  c +  304 

-24e 

-160 

3 

18 

24 

36 

^420 

-2 

-12 

-    16 

-    24 

280 

6 

8 

12 

-140; 

28e-140 

-24c 

+  120 

We  thus  obtain  the  same  remainder  as  by  the  former 
method,  and  consequently  the  same  result.  A  comparison 
of  the  two  methods  shows  that  they  are  but  slightly  differ- 
ent in  form,  but  the  second  method  shows  rather  more 
clearly  that  c  need  not  be  introduced  into  the  dividend  at 
all,  but  the  proper  multiples  of  it  found  by  the  preliminary 
reduction  can  be  added  to  or  taken  from  the  numerical 
remainder,  and  the  "true  remainder"  be  thus  found,  and 
c  determined  from  it. 


Find  the  value  of : 


Ex.  20. 


1.  x^ —  3x^-{-4:x'^—3x  +  4:,  given  x^  =  x~l. 

2.  x'  -  2x'  -  Ax'  +  l^x'  -  11:^'   -  10,  given  (x-  If  =  2. 


68  THEORY    OF   DIVISORS. 

3.    2x'-~7x'+12x'-llx'  +  2x-5,  given  (x-iy +  2^0. 

given  x^'}-Sx'^—2x-\-  5  =  0. 

5.  6:?;'  + 9^^-16:?;* -5:^^ -12:^^-6^ +  60, 

given  3x^  -^  x  —  A  =  0. 

"What  values  of  c  will   make   the   following   polynome 
vanish  under  the  given  conditions: 

6.  :^*+ 13^^^  +  26:^2  +  52:^ +  8^,  given  :?;  + 11  =  0. 

7.  x'  -  2x'  -  dx'  +  2cx  -  14,  given  3:r  +  7  =  0. 

8.  X*  —  4:X^  —  x"^ -{~16x-\-6c,  giyen  x"^  =  x-\~6. 

9.  2:^*— 10rr'  +  4c:?:+6,  given:i:'  +  3  =  3:r. 

10.  2x'  +  x^—7cx^+llx+10,  given  2x  =  6. 

11.  4:i'*  +  c?:r'  +  110:?;-105,  given  2:^'- 5:r+ 15 -- 0. 

12.  3:r^-16:i;*  +  e:z;'-5:i;'-^114:^;+200,  given  :i;'  =  3.'?;-4. 

13.  What  values  of  ^  and  q  will  make 

x^-j-  2x^-—10x^—px-{-  q  vanish,  if  x'^ —  S(x  —  1)? 

14.  What  values  of  ^  and  q  will  make 

a}'  -  5  a^«  +  10  a^  -  15  a'  +  29  a'  -^^  pa'  +  q  vanish,  if 
(a'-2y  =  a''~S? 

Theorem  II.    If  the  polynome  f(xy  vanish  on  substi- 
tuting for  X  each  of  the  n  (different)  values  ai,  a^,  a^, ,  a,„, 

f{xy  —  A(x~  ai)  (x  —  a^  (x  —  a^) (x  —  a^), 

in  which  A  is  independent  of  x,  and  consequently  is  the  co- 
efficient of  x"^  inf(xy. 

Cor.    If  /(xy  and  (j>  (x^  both  vanish  for  the  same  m  dif- 
ferent values  of  Xjf^xy  is  algebraically  divisible  by  <^0O"*- 


THEORY   OF   DIVISORS. 


59 


Examples. 

x^  +  ax'^  -]-bx-}-c  will  vanish  if  2,  or  3,  or  —4  be  sub- 
stituted for  X ;  determine  a,  b,  c. 
The  coefficient  of  the  highest  power  of  a;  is  1  ; 
.-. :(?  +  ax"  -^lx-\-c-=-{x-~2)  (x  —  3)  (x  +  4) 

.-.^--1;  h  =  ~U;  0  =  24:. 

x^  -\- hx^  -{- ex -\-  d  will  vanish  if  —  3,  or  2,  or  5  be  sub- 
stituted for  X ;  determine  its  value  if  3  be  substituted 
for  X. 
The  given  polynome  =  (a;  +  3)  (x  —  2)(x  —  b)\ 
:.  the  required  value  is  (3  +  3)  (3  -  2)  (3  -  5)  =  -  12. 


^     3.    aa^ 


3  hx'^  -\- 2>  ex -\-  d  will  vanish  if  for  x  be  substituted 
—  3,  or  |-,  or  1|-,  but  it  becomes  45  if  for  x  there  be 
substituted  3  ;  determine  the  values  of  a,  &,  c,  d. 
The  coefficient  of  the  highest  power  of  a;  is  a ; 

H); 


H); 


If  01?  -\-jpx^  -\~  qx-{-r  vanish  for  x^^a,  or  h,  or  c,  deter- 
mine p,  q,  and  r  in  terms  of  a,  ^,  c. 
rz;^  +p:?;^  +  qx-\-r  =  {x  —  a)  (^'  —  5)  (a;  —  e) 

=^  x^  —  {a -\- h  '\- c)  x'^  -\-  {ah  -\-hc-\-  ca)  x  —  abc. 
:.p=^  —  (a  +  b  +  c)  or  —  5a, 
^  =  a5  +  &c  +  ea    or  5  a^, 
r  =  —  abc  or  —  5  abc. 


: .  aa?  +  ?,hx'  +  2,cx  +  d=  a{x  +Z){x - 
.•.a(3  +  3)(3^i)(3~li)  =  45; 

•    ^y  —  9 

-i)(^ 

.-.  2x^  +  Zbx^  +  Zcx  +  cl^2{x-\-?,){x- 
,•,6  =  1;  c  =  -3i;  £?=4f 

--i)(^ 

60  THEOEY    OF   DIVISORS. 

5.  If  x^  -{-px^  -\-  qx  -\-r  vanish  for  x  =  a^  ot  h,  or  c,  deter- 
mine the  polynome  that  will  vanish  for  x  =  h  -\-  c^ 
or  c  +  a,  or  a  +  S. 

Since  x^  -\-px'^  -\-  qx-\-T  vanishes  for  a;  =  a,  or  h,  or  ^, 
x^  -—px^  -\-  qx  —  r  will  vanish  for  x=^  —  a,  or  ~  h, 
or  —  c,  and  — p  =^  a-{-h  -{-  c. 

But  the  required  polynome  will  vanish  for 

X  =  —p  —  a,  or  —  p  —  Z),  or  ~p  —  c  ; 

that  is,  for  x-{-p  =  —  a,  or  —  b,  or  —c. 

Hence,  it  is  (x  -{-j^T  ~~P  (^  +J^)'^  +  ^  (^  +i^)  —  ^' 
=  ^^  +  2^9^^  +  (p'^  -\-  q)x  +pq  — '?'. 

The  following  is  the  calculation  in  the  last  reduction. 
(See  page  37.) 


-p 

1 

7' 

p 

0 

!z; 

pq-r 

p 

p\ 

f+q 

p 

1 '' 

2p 

6.  In  any  triangle,  the  square  of  the  area  expressed  in 
terms  of  the  lengths  of  the  sides,  is  a  polynome  of 
four  dimensions ;  and  the  area  of  the  triangle,  the 
lengths  of  whose  sides  are  3,  4,  and  5,  respectively, 
is  6.  Find  the  polynome  expressing  the  square  of 
the  area. 
Let  a,  h,  and  c  be  the  lengths  of  the  sides,  and  A  the 

required  polynome. 
1.    The  area  vanishes  if  any  two  of  the  sides  become  to- 
gether equal  to  the  third  side  ;  hence,  if  a  -f-  Z)  =  <?, 
A=^0,  and  consequently  A  is  divisible  by  a-\-h  —  c. 
Similarly  it  is  divisible  hj  h-\-c  —a  and  hy  c-\-a  —  h. 


THEORY    OF    DIVISORS.  61 

2.    The  area  vanishes  if  the  three  sides  vanish  together ; 

hence,  if  a  +  Z)  +  c  =  0,  A  =  0,  and  consequently  A 

is  divisible  by  «  +  &  +  <?. 
We  have  thus  found  four  linear  factors,  but  A  is  of  only 

four  dimensions. 

.-.  A=m(a+h  +  c)(h-\-c  —  a)(c-{-a  —  b)(a+b  —  c), 

in  which  m  is  a  numerical  constant. 

But  6^  or  36  =  m(3+4+5)(4+5-3)(5+3'  4)(3+4-  5) 
=  576  m;  .'.  m  =  -^. 

The  above  includes  all  the  ways  in  which  the  area  of  a 
triangle  can  vanish,  for  the  vanishing  of  only  one  side  in- 
volves the  equality  of  the  other  two  ;  or,  if  a  =  0,  ^  =  c, 
and  therefore  a-\~b  =  c,  which  is  included  in  (1).  If  two 
sides  vanish  simultaneously,  the  three  must  vanish. 

Examples  on  the  Corollary. 

7.  Prove  that  (x  +  lf'-x'''  ~2x~\  is  divisible  by  2^' 

Factoring  2^^ -^^x^ ■\-x,  we  find  it  vanishes  for  rr  =  0, 
or  —  1,  or  — -|-.  Substituting  these  values  in  the  first 
polynome,  it  also  vanishes.  But  these  are  different 
values  of  x,  hence  the  truth  of  the  proposition. 

8.  {x -\- y -\-zf —  x^ —  y"  —  z"  is  divisible  by  {x-\-y-^zf 

—  x?  —  y^  —  z^. 
The  last  expression  vanishes  ii  x^=—y,  so  also  does  the 

first. 
By  symmetry  they  both  vanish  if  y^—z  and  if  z^^—x. 

Hence,  they  are  both  divisible  by  {x-\-y){y-\-z){z-\-^). 

But  this  expression  is  of  three  dimensions,  as  also  is 

the  second   of  the    given   polynomes,  hence  it  is  a 

divisor  of  the  former. 


62  THEORY    OF   DIVISORS. 


Prove  that 

+  [{b  +  cf+{a  +  df]{b~c){a-d) 

+  [{b  +  df  +  {c  +  af]{b-d){c-a) 

is  algebraically  divisible  by 

{a-h)  (c-d)  (b-c)  (a~d)  (b-d)  (c~~a)(a+b+c+d), 

and  find  the  quotient. 
Let  a  — -  h,  and  the  first  polynome  reduces  to 
[{a  +  cf  +  (a  +  df]  (a  -  c)  (a  -  d) 

+  [{a  +  dy+  (c  +  ay]  (a  -d){c-  a) 

which  vanishes,  the  second  complex  term  differing 

from  the  first  only  in  the  sign  of  one  factor,  having 

(c  —  a)  instead  of  (a  —  c). 
Hence,  the  first  polynome  is  divisible  by  a  —  b,  and 

by  symmetry  it  is  also  divisible  hj  a  —  c^hj  a  —  d, 

hj  b  —  c,  hj  b  —  d,  hj  c  —  d. 
Again ,  (a + bf + (<? + d)^  is  divisible  by  (a-\-b)  +  {c+d)', 

for,  on  putting  a-{-b^  —  (c-]-  d),  it  becomes 
■     [-(c  +  d)Y  +  ic  +  df==0. 
Similarly  the  other  terms  of  the  first  of  the  given  poly- 

nomes  are    each    divisible   by  a  -}-  b  ~{-  c  -{-  d,  and 

consequently  the  whole  is  so  divisible. 
Now  all  these  factors  are  different  from  each  other, 

hence  the  first  of  the  given  polynomes  is  divisible 

by  the  product  of  these  factors ;  that  is,  by  the  sec- 
ond of  the  given  polynomes. 
Both   of  these   polynomes   are   of  seven  dimensions, 

hence  their  quotient  must  be  a  number  the  same  for 

all  values  oi  a,  b,  c,  d. 
Put  a  =  2,  b  =  l,c  =  0,  d  =  -l,  and   divide.      The 

quotient  will  be  found  to  be  —  5. 

.■.[(a+  bf+(c  +  dy](a-b){c-d) 

+  [(i  +  of  +  (a+df] (b  -c)(a-  d) 

+  [{b  +  dy'+{c  +  of]  (b  -d)(e~  a)  = 

-5(a~b)(c~d)(b'C)(a-d){b-d)(f-aXa+b+c+d). 


I 


THEORY    OF    DIVISORS.  63 

Note.    It  is  not  always  necessary  to  find  the  factors  of  the  divisor, 
as  the  following  examples  show. 

10.    Prove  that  ^^  +  :r  +  1  is  a  factor  of  x^^  +  rr'  +  1. 

x^  -^  x-\~l  will  be  a  factor  of  x^^  ~{~  x"^  ~\-  1^  provided 
x''  +  x'  +  l  =  0,i^x'  +  x+l  =  0. 

/.x^  +  x''  +  x  =  0, 
.\x'  +  x'  +  x+l  =  l, 

.*.  ^^  =  1  and  x^^  =  1, 

.•.  x"^  =  X  and  x^^  =  x^, 

.-.  x''  +  x'  +  l  =  x'  +  x+l  =  0, 

.'.  ^^^  +  ^  +  1  is  a  factor  of  x^^  +  x'  +  1- 

Two  other  methods  of  proving  this  proposition  are  worthy 
of  notice. 

I.    x'^  -{-  x-{~l  will  be  a  factor  of  x^^  +  .r^  +  1,  provided 
it  is  a  factor  of 

[(x''  +  :r^  +  1)  ±  a  multiple  of  (rt^=^  +  x+  1)]. 
x^^  -{-x"^  ~\-l  differs  by  a  multiple  of  x"^  +  •'^  +  1  from 
x''  +  x''  (x'  +x  +  l)  +  x\x''  +  x  +  l)  +  x' 

.  +x'  (x'  +  x+l)+x(x'  +  X  +  l)  +  l 
=  x''  (x'  -{-x+l)  +  x'  (x'  +  X  +  1)  +  x'  (x'  +  X  +  1) 

+  x\x''  +  x+l)  +  (x'  +  X  +  1) 
=  (x''  +  x'  +  x'  +  x'  +  1)  (x'  +  X  +  1). 
Hence,  x^  +  .r  +  1  is  a  factor  of  x^^  -f-  x"^  +  1. 

jj     x''  +  x'  +  l^y~l  ^x-l 
x'  +  x+l        x'~l       x'-l 
_  (x''  -  1)  \(x''  -l)-x  (x''  -  1)1 

{x'-l){x'-l) 
^  (x''  -  1)  (x''  ~  1)      X  (x''  -  1)  (x''  -  1) 
(x'  -  1)  {x'  -  1)  (x'  -  1)  (x'  -  1)  * 


64  THEORY    OF    DIVISOES. 

But  we  see  at  once  that  on  reduction  both  of  these 
fractions  give  an  integral  quotient ;  hence, 

(x^^  -\- x^  ~\- 1)  -^  x^  -{- X  ~\- 1  gives  an  integral  quotient. 

11.    r^^  +  a;  +  1  is  a  factor  of  (x  +  1)^  —  x'^  —  1. 

If  x'^-{-x  -\~1  =  0,  (x-\-iy  —  x^  —  1  will  vanish  also  ; 
for  in  such  case  x  -\~  1  ^= -—  x"^ . 

.-.  (x  +  1)'  --x'-l  =  {-~  xj  -x'-l  =  -x''-x'-l, 
which  by  the  last  example  vanishes  if  x'^-{-x-{-l=^0. 

/ .  x'^  -}- X -{- 1  is  Si  factor  of  (x  +  ly  —  x"^  —  1. 

For  X  substitute  _,  and  multiply  by  y^  and  y^,  respec- 
tively,  and  this  example  becomes 

x'^  -{~xy-\~  y^  is  a  factor  of  {x  +  yj  —  x^  —  y^ 

Ex.  21. 

Determine  the  values  of  a,  h,  c,  d,  e,  in  the  following 
cases : 

1.  x^  -\-?>  hx^  -^?icx  -^  d  vanishes  for  :^  =  2,  or  3,  or  4. 

2.  x^  -\-  cx^  -\-  dx  ^  e  vanishes  for  x  =  \\,  or  —  3,  or  \\. 

3.  x^  -\-  hx^  +  (^^  +  24  vanishes  for  x  —.%  or  —  3. 

4.  ax^  +  hx^  +  c:r  +  90  vanishes  for  ^  =  3,  or  —  5,  or  2. 

5.  ax'^  -\-  cx^  —  ^^x  -\-  e  vanishes  for  x  =  \\,  or  —  4,  or  2|-. 

6.  ^\x^-\-^cx^-^^dx^e  vanishes  for  x  =  1 1,  or  —  3-|-,  or  1-^. 

7.  ax^  +  ^^^  +  G-^^  —  81  vanishes  for  x  —  f ,  or  |,  or  3. 

8.  ax^-\-  cx^-\-  dx  -\-  e  vanishes  for  ;r  =^  2,  or  \\,  or  —1,  and 

becomes  14  for  :r  =:  1. 

9.  ax^  -\-cx-\-d  vanishes  for  x  ==  1^1,  or  2f ,  and  becomes  49 

for  57  —  3,  determine  its  value  for  rr  —  —  3. 


THEORY    OF    DIVISORS. 


65 


Given  that  x^~px'^-\'qx  —  r  vanishes  for  x^a,  or  b^  or  c, 
determine  the  polynome  that  vanishes  for  : 

10.  :?;  =  a+ 1,  or  ^  +  1,  or  (?  + 1. 

11.  ^  =  a— 1,  or  ^  —  1,  or  <?— 1. 

12.  x  =  l ,  or  1  —  -,  or  1 

a  0  c 

13.  x  =  ah,  or  he,  or  ca. 

14.  x^=  o?,  or  Z>^,  or  cl 

15.  X  ==  a{h  +  <?),  or  h'{c-^  a),  or  c(a  +  h). 


I    [ 


a(h  +  c)-=q--^ 


16.    X-- 


a-\-h         h-\-c         c- 
:  — ! — ,  or  — ■ — ,  or  — 


fa  +  h^p      iV 


Prove  that  the  following  are  cases  of  exact  division  : 

17.  {x  -  ly  -  x^  +{x'  -  X  +Vf  ^  X?  -  2x'  +  2:^-1. 

18.  {x -ly - x^+  {x" - X -\-lf ^ x^ -2x'  +  2x -I 

19.  {x  -  2y(2x  -  by  -  x''  +  2''(x'  -4^  +  5)^ 

-^x'-Qx'  +  lSx-lO. 

20.  (x'  +  4:x  +  Sy^-  x'^ -  x'  -  5x  -  ^-i-x^  +  ex'  +  Sx+S. 

21.  (9x-4:y'(x-iy' - x^^-^^x" -  Ux  +  4)'^ 

-r-(x-l)(9x-4:)(9x'~14:X  +  4:). 

22.  [6 (x  -  1)] ''  -  (2x'  +  Sx-4:y'  +  (2x'  -Sx  +  2^ 

-^(2x'  +  Sx-4:)(2x'-Sx  +  2)(x-l). 

23.  [2(x+l)(x-2)y'+{x'-3x+Sy'-(^x'-bx-lf 

~-  (x  +  l)(x  -  2)(x'  -^x  +  S)(3x''  -  5x-l). 

24.  [6(x~l)y'-(2x'  +  Sx-  4)^«  -(2x''-Sx  +  2^ 

+  2(22;^  +  3^  -  4)«  (2:z;2  -  3^  +  2)^* 
H-(2;-l)(2^'  +  3ri:-4)(2^^-3^  +  2). 


66  THEORY    OF    DIVISORS. 

25.  [2(x  +  1) (x-2)Y'  -  (x'  -  Sx  +  Sy'-(Sx'-bx-iy' 

+  2(ix'-Sx  +  Sy  (Sx'  -  5x  -  ly' 

-^(x  +  l)(x--  2) (x'  -3^  +  3) (3^-^  -  bx  -  1). 

26.  1  +x'  +  x^^l  +  x  +  x\ 

27.  x'''  +  x^f  +  y'''-^x''  +  X7/  +  y\ 

28.  l  +  a^  +  x^  +  x^  +  x^'-^l+x  +  x^  +  a^  +  x'. 

29.  l+x'  +  x'  +  x'''  +  x^^-^l  +  x  +  x''  +  x'  +  x\ 

30.  x^^  +  ;r'V  +  ^y'  +  /^  -^  ^^  +  ^r'y  +  ^y'  +  y^. 

31.  ^"  +  a;*  +  :2;'  +  :z;+l-^a;*  +  ^'  +  ^'  +  :?;  +  l. 

32.  l+x  +  x'  +  x^  +  x'^  +  x^  +  x^ 

-^l  +  x  +  x'  +  x^  +  x^  +  x''  +  x\ 

Find  the  quotient  of  the  following  divisions,  in  which  D 
denotes  the  product : 

{b  -  c){c -  a){a-h){a  -  d){h  ~  d){c -  d). 

33.  (h'c^+a'd''){b-c){a-d)  +  (c'a''+h''d''){c~a){h-d) 

+  (a'b'  +  c'd')  (a  -b)(c-d)-^  B. 

34.  (bc+ad){b'-c')(d'-d')  +  (ca  +  bd)(c'-a')(b'-d') 

+  (ab  +  cd)  (a'  -  b')  (c'  -  d')  ^  D. 

35 .  {b-\-c){a^d) (IP' — c^) (a^  —  d'^)  +  the  two  similar  terms 

36.  (IP  +  c")  ip?  +  d"")  (b  -c)(a-d)  +  the   two   similar 

terms  -^  D. 

37.  [be  (b  +  cy  +  ad(a  +  dy]  (b  -  c)  {a  -  d)  +  the  two 

similar  terms  ^  D. 

38.  [bc{b  +  c)  +  adia  +  cZ)]  {b''  -  &)  (a^  -  d'')  +  the  two 

similar  terms  -^  2>. 

39.  [5c(Z>^  +  c^)  +  acZ(a^  +  (^0](^-6?)(a-c^)  +  the  two 

similar  terms  -j-  D. 


I  THEORY    OF    DIVISORS.  67 

40.  (h  -\~  c  —  a  —  dy(b  ~  c)  (a—d)  +  the  two  similar  terms 

41.  The  sum  of  the  fractions  \,  |-,  |-,  ,  ^,  increased  by 

the  sum  of  their  products,  two  by  two,  increased  by 

»the  sum  of  their  products,  three  by  three,  ,  in- 
creased by  their  product  is  equal  to  n. 

42.  In  any  trapezium,  the  square  of  the  area  expressed  in 

I  terms  of  the  lengths  of  the  parallel  sides  and  the 

diagonals,  is  a  polynome  of  four  dimensions ;  deter- 
mine that  polynome. 

43.  In  any  quadrilateral  inscribed  in  a  circle,  the  square 

of  the  area,  expressed  in  terms  of  the  lengths  of  the 
K  sides,  is  a  polynome  of  four  dimensions ;  find  that 

K  polynome. 

Theorem  III.  If  the  polynome  /(^)"  vanish  for  more 
than  n  different  values  of  x,  it  vanishes  identically,  the  co- 
efficient of  every  term  being  zero. 

Co7\  If  a  rational  integral  expression  of  ?^  dimensions  be 
divisible  by  more  than  n  linear  factors,  the  expression  is 
identically  zero. 

Examples. 

^     (x  —  a)(x  —  h)(x  —  b)(x  —  c)(x~c)(x~a)     -i  __n. 
^  *    lc~a)(c-b)'^(a-b)(a-c)'^(b-~c)(b-a)~  ' 

if  a,  b,  and  c  are  unequal ;  for  this  is  a  polynome 
of  two  dimensions  in  x,  but  it  vanishes  for  x  =  a, 
and,  therefore,  by  symmetry,  hi  x  =  b,  and  for  x^=  c; 
that  is,  for  three  different  values  of  x ;  hence,  it  van- 
ishes identically. 

[(a  +  by  +  (c  +  dy](a-b)(c-d) 
+  [(b+  cy+(a  +  df]  (b-c)(a-d) 
+  [(o  +  ay+  (b  +  dj]  (c  -a)(b-d)^  0. 


68  THEORY    OF    DIVISORS. 

Substitute  b  for  a,  and  tlie  expression  becomes 

[(h  +  cy  +  {h+dy]{b-c){h~d) 
+[{c+hy+(h+ciy]{c~h){h-d), 

which  vanishes;  hence,  the  given  expression  is  divi- 
sible by  a~h,  and  consequently,  by  symmetry,  it  is 
divisible  by  (a  — &),  (5  — c),  {c  —  d),  (a  —  c),  {b  —  d), 
and  (a  —  d).  But  the  given  expression  is  of  only 
four  dimensions,  while  it  appears  to  have  six  linear 
factors  ;  hence,  it  vanishes  identically. 

Ex.  22. 

Verify  the  following : 

b'c'  b'(b'-/)  ^  c\c'~b') 

=  x'  +  y''  +  z''-b'-c\ 

b'c'  b\h'-c')  c'{c'~b') 

3     x'y'      (x'-d')(a'--y')z'  .   (x' -  b')  (b' ~  y')  z' 
a'  b''  ~^  if  -  a^)  (Z)^  -  a')  a"  "^  (&^  -  z')  (b'  -  a'')  W 

~^  {z^-a^){W'-z^) 

1  ,  1 


4. 


(^  +  a)(a-Z))(a  — c)      (x-\-b){b  —  c){b-a) 
1  1 


+ 


{x ~\-  c){c  —  a)  (c  —  b)      (x-{-  a) (x  -{-b)(x  -\-  c) 

5 .    be  (b'  -  c')  +  ca  (c'  -  a')  +  ab  (a'  -  b') 

=  (a  +  b  +  c)[a\b  -  c)  +  .b''(c  -  a)  +  c'(a-b)l 

6  <^  +  ^  I  ^  +  y  I  <^  +  ^ 

^(^  —  y)(x-z)      y(y-x)(y-z)      z{z-x){z  —  y) 

a 

'^  xyz 


THEORY    OF    DIVISORS.  69 

[^     a'  (b'  -  c')  +  h'  (c'  -  a')  4-  c'  (a'  -  b') 
d'{b-~c)  +  b\c~a)  +  c\a~b) 
=.^[(a  +  b+cy-a'-  b' - c'l 

8.    (adf-\-  bcf-\-  bed  —  acc)^  +  (bee  +  aed  +  aef —  bdfy 

(a  —  b)(b  —  c)  (c  —  a) 

^^[{a-by  +  ib-^ey  +  ic-d)-^]. 

10.    {~x  +  y  +  z){x~y  +  z)(x  +  y  —  z) 

+  x{x  —  y  +  z)(x+y  —  z)+y{x+y  —  z){~x  +  y+z) 
+  z{-~  X  +  y  +  z){:x  —  y  +  z)  =-  4:xyz. 

^     {a'  -  by  +  (^'  -  cy  +  (g^  -  g^)^ 
{a  +  b)(b  +  e){e  +  a) 
_  (^  _  Z,)3  +  (^  _  ^y  +  {c~  d)\ 

12.    :^^(y  +  ^7  +  y'^(2  +  ^-)'  +  ^'(^'  +  y/  +  2^yK.t;  +  y  +  ^) 
=^2(xy+yz  +  zx)\ 


I 


Theorem  IV.  If  the  polynomes  f{xY,  ^{xy  (n  not 
less  than  m),  are  equal  for  more  than  n  different  values  of 
Xj  they  are  equal  for  all  values,  and  the  coefficients  of  equal 
powers  of  x  in  each  are  equal  to  one  another. 

This  is  called  the  Principle  of  Indeterminate  Coefficients. 

Examples. 

1.    ^l + ^1 

(a  -  b){a  -c)(a~d)      {b~^  a){b  -  c)(b  -  d) 

-I ^ y- =  0. 

•  (c  —  d)(G~h)(c  —  d)      (d—a)(d—b){d—c) 

{x  ~  a)  (x  —  b){x  —  c)  (x  —  d) 
A      ^     B     ^      C     ^      D  /  X 

x —  a      X  —  0      X  —  c      X  —  a 
in  which  A,  B,  (7,  D,  are  independent  of  x. 


70  THEORY    OF    DIVISORS. 

Multiply  by  (x  —  a)  (x  —  h){x-  c)  {x  —  d). 

.'.  x"^  =^{A  +  B  -\-  C-\~  D)x^  +  terms  in  lower  powers 
of  x'. 

Now  this  equality  holds  for  more  than  three  values  of 
X,  holding  in  fact  for  all  finite  values  of  x. 

,'.A  +  jB+C+D  =  0.  (/3) 

Again,  multiply  both  sides  of  (a)  by  x~  a, 

^ 

(x  —  b)(x  —  c)  (x  ~  d) 

\x  —  o      X  —  c      X  —  a  J 
Put  x  =  a, 

-J. 

(a  —  h){a~  c)  (a  —  d) 

By  symmetry, — — -jr^B,  etc. 

(6  —  a){b~  c) (6  —  d) 

Adding, 

(I '^ 

{a~b){a-  c){a~  dy  {h  ~  a){b  -  c){b  -  d) 

+ ^1 + ^- - 

{c-a){c~b){c-d)      {d-  a)  {d  -  b)  {d  -  G) 

=  A-^B-\-C+D  =  ^,  hj(P).  . 

2     a\a-\-b){a  +  c)      b\b -\- e){J> -\~  a)      c\c  +  d){c-\-b) 
{a-~b){a-c)  "^    {b-c){b-a)'^    (c-a)(c~b) 
=  (a  +  b  +  cy. 

Assume  x^  —  px"^  -{-  qx  —  r  =^  {x — a){x—b)(x—c)     (a) 

.-.  x^  +px^  +  qx  +  r  =  {x  +  a){x ~\-  b) {x  +  c)  (^) 

x^JrP^  +  q:>?  +  r.^  _A_        B       ^    ^  ^ 

ar—px^+qx  —  r  ^      x—a     x—b      x—c 


THEORY    OF    DIVISORS.  71 

Multiply  by  x^  —  px^  -\-  qx  —  r,  and  equate  tlie  coeffi- 
cients of  the  terms  in  x^.  In  multiplying  the  frac- 
tions in  the  right-hand  member  of  (y),  use  the  factor 
side  of  (a). 

:.A  +  B+C=2p\ 

Multiply  both  members  of  (y)  by  x  —  a. 

x^x+d)  {x-]-h)  (x-{-c) 
{x—b)(x  —  c) 

=  ^  +  (.  +  2^  +  ^  +  ^)(.-a). 
Put  x^=aj 

(a  ~  b)(a~  c) 
By  symmetry, 

2¥{b  +  c){b  +  a)  p        ,   %c^c  +  a){c  +  b)_^ 

{b-c){b-a)  '              (c-a)(c~b) 

'    .  a'(a  +  b)(a  +  c}  .  b'(b  +  c)(b+a) 

"    (a-b)la~c)  (b-c)(b-a) 

(c  —  a)  (c  —  o) 
=  (a  +  b  +  cy. 

3.    Extract  the  square  root  of  1  +  ^  +  ^"^  +  ^^  +  ^*  + 

Assume  the  square  root  to  be 

1 -}-  ax  ~\- bx"^  -\-  cx^  -f  dx^  + 

/.l  +  x  +  x^  +  x'+x'4- 

-=-(l  +  ax  +  bx'  +  cx^  +  dx'  + J 

=  1  +  "Aax  -f  {a^  +  '2.b)x^  +  2{ab  +  g)x^ 

+  (2c^+2ac+Z)')a;*+ 


THEORY    OF    DIVISORS. 


.■.2a  =  l, 

.■.«  =  i 

2b  +  a^  =  l, 

*=i(l-i)  =  f. 

2(c  +  ab)  =  l, 

«=i-(iXf)  =  T^, 

2di-2ac  +  b'=l, 

<^=i(l-A-A)  = 

-  35 

-  128- 

.■.^{l+x  +  x'  +  -- 

•)  = 

Note.  As  it  is  frequently  necessary  to  determine  the  coefficient  of 
a  particular  power  of  x,  a  few  preliminary  exercises  are  given  on  this 
subject. 

Ex.  23. 

Determine  the  coefficient  of: 

1 .  ^*  in  (1  +  axf  +  (1  +  bxf  +  (1  -  cxf, 

2.  x'm(li-x  +  2x'  +  3x')(l  —  x  +  Sx''  +  x^~5x'),      ' 

3.  x'm(l  +  x  +  2x'  +  Sx^  +  4:x'  + ) 

(l-x  +  x^-x^+x'  — ). 

4.  x"^  in  A{x  —  h)  {x  —  c) (x  —  d)  +  B (x~ a) (x  —  c) (x  —  d) 

+  C(x~  a){x—  h)  (x  —  d)  +  D {x  —  a)  (x—h)  {x— c) 

5.  :^*  in  (1  —  axY  (1  +  axj*, 

6.  x^in{l  +  axy(l  —  hxf, 

7.  In  the  product 

(1  +  ax -\- hx^  -\-  cx^  -\- )  (1  —  ax -{- hx'^  ~  cx^  + ) 

prove  that  the  coefficients  of  the  odd  powers  of  x 
must  be  all  zeros. 

Determine  the  value  of  the  following  expressions : 
1  +  ^ 


(a-b)(a-c)(a-d)      {h  -  a){h  -  c)(b- d) 

{c-  a){c  -  b)(c-d)  ^  (d-  a)(d-b)(d-  c) 


THEORY    OF   DIVISORS.  73 


I 


"^ + ^^ + 

{a~h){a-c){a-d)      (b-a)(h-c)(b-d) 


10. — — — —  +  three  similar  terms. 

{a  ~b){a—  c)  (a  —  a) 

11.  ■; =-— — —  +  three  similar  terms. 

{a  —  o)(a  —  c)  (a  —  a) 

12.  ; -— — —  +  three  similar  terms. 

(a  —  b){a  —  c)  (a  —  a) 

13. — — — -  +  three  similar  terms. 

(a  --  o)(a  —  c) {a  —  a) 

14.  — > -^ -^  -(-  two  similar  terms. 

(a  —  D){a  —  c) 


15  •    -7^ Y~-r ^  +  two  similar  terms. 

{a-b)i^a-c) 

W        a\a  +  h){a+c)   .    .         ••14- 

16.  ^        // -^  +  two  similar  terms. 

{a  —  o){a  —  c) 

17.  <a  +  ma  +  c)(a+d)  ^  ^^^^^  ^.^.^^^  ^^^^^ 

1^  (a  —  0)  {a  -—  c){a  —  a) 

18.  V    "T"   A    ~^  J^       7x    +  three  similar  terms, 
(a  ~  b){a  —  c)  {a  —  a) 

19.  ay  +  h){a  +  c){a  +  d)      ^^^^^  ^.^.^^^  ^^^^^ 

(a  —  b){a  —  c){a—  a) 

20. ;,7"        N  +  two  similar  terms. 

(a  ~b){a  —  c) 

For  numerator  use  px^  —p^x  -{-pq  —  r. 

21.  ^- =4-^ — ^  +  two  similar  terms. 

(a  —-  b){a  —  c) 

For  numerator  use  3  x^  -{-px  +  q. 


74  THEORY    OF    DIVISORS. 

22. ^-— - — I — -  4-  two  similar  terms. 

(a  —  b)(a  —  c) 

For  numerator  use  x(x—p). 

23.  ; ,,  /~ — '^^ +  three  similar  terms. 

(a  ~  b)(a  —  c) {a  —  a) 

24. \^  T* —, — ^— -  +  three  similar  terms. 

{a  —  b){a  —  c)  (a  —  a) 

25. — i^ '\- -^  +  three  similar  terms. 

{a  —  o)(a  —  c)  {a  —  a) 

Extract  the  square  root  to  four  terms : 

26.  l  +  x.         27.    l-x.         28.   l  +  2x  +  ?>x'  +  4:a^+ ^ 

29.  1  -  4^  +  10^'  -  20 x^  +  35^*-  56^^  +  S^x\ 

30.  Extract  the  cube  root  of  1  +  ^  to  four  terms. 


§  11.  1.  Find  the  condition  that  px^  -{-  2qx  -\-  r  and 
^V  -[-2q^x-\-  r'  shall  have  a  common  factor. 

Multiply  the  polynomials  by  p^  and  p  respectively,  and 
take  the  difference  of  the  products ;  also,  by  r'  and  7%  re- 
spectively, and  divide  the  difference  of  the  products  by  x. 


p^px^  -f  2p^qx  -{-pW 
pp^x^  +  2pq^x  -{-pr^ 


pr^x^  -\-  2  qr^x  +  rr' 
pWx^  ■j-2q'rx-\-  r'r 


2  (pq^  —p'^)  ^  +  (P'^'  —p^r) 


(pr'  — j9  V)  x-j-2  (qr'  —  r^q) 


Multiply  the  former  of  these  remainders  by  (pr^—p^r),  and 
the  latter  by  2(pq'~p'q),  and  the  difference  of  the  products 

(^r'  —pWy-  —  4  {pq^  ~pW)  (S'^'  —  '^'$')- 

But  if  the  given  polynomials  have  a  linear  factor,  this 
remainder  must  vanish,  or 

{pr^  —  pWy  =^  4:(pq^  --  p^q)  (qr^  —  r^q) . 


THEORY    OF    DIVISORS.  75 

If  the  given  polynomes  have  a  quadratic  factor,  the 
linear  remainders  must  vanish  identically,  or  (Th.  III.), 

pq^  -p^q  =  0,  pr^  —p^r  =  0,  and  qr'  ~  r^q  =  0, 

or         P  =  9^  =  L 
p       q^      T 

2.  Find  the  condition  that  px^-^Zqx^A^^rx^s  shall  have 
a  square  factor. 

Assume  the  square  factor  to  be  {x  —  (if.  On  division, 
the  remainder  must  be  zero  for  every  finite  value  of  x,  and 
consequently  (Th.  III.)  the  coefficient  of  each  term  of  the 
remainder  must  be  zero.  Divide  by  (x  —  (if,  neglecting 
the  first  remainder. 


a 

p           ^q 

pa 

Sr                 s 

po^  +  %qa 

a 

p       pa  +  Sq 
pa 

2pa^  +  ?>qa 

p     2pa  +  3q] 

3(pa^  +  2ga  +  r) 

.'.pa^  +  2qa  +  r  =  0] 

.'.  px'^  ■j-2qx-\-r  is  divisible  by  ^  —  a  ;  (Th.  I.  Cor.  2.) 
or,  px^  +  3  qx"^  -}-Srx-{-  s  and  px"^  -\-2qx-{~  r  have  a  common 
divisor.  Multiply  the  latter  polynome  by  x,  and  subtract 
the  product  from  the  former,  and  the  proposition  reduces  to 

lfpx^-{-Sqx'^-\-Srx-{-s  have  a  square  factor,  ^.^^  +  23':^+^ 
and  qx'^-\-2rx-{-s  will  have  the  square  root  of  that  factor 
for  a  common  divisor. 

Ex.  24. 

1.    Determine  the  condition  necessary  in  order  that 

x^-^-px  -{-  q  and   x^  ~{-p'x-\-  q^  may  have  a  common 
divisor. 


76  THEOEY    OF    DIVISORS. 

2.  The  expression  x^  +  ^a^ r'+Zhx'  +  cx^+Mx'  +  '^e^ x+f 

will  be  a  complete  cube  if 

'r     e      d      c  —  a^      i        A  M 

-^      a      h        6a'  I 

3.  Prove  that  ax^-\-  hx-{-  c  and  a-\-hx'^  -\~  ex"  will  bave  a 

common  quadratic  factor  if 

y  c"  =  {c'  -a'+  b')  (c'  -  a^  +  ah). 

4.  Prove  tbat  ax^  -{-ha^-]-c  and  a  +  ho^-j-  ex"  will  have  a 

common  quadratic  factor  if 
a'})'  =  {a'  -  c''){a'  -  c'  -\-lc). 

5 .  Prove  tbat  ax^ -^-hx^ -\- ex -\- d  and  a  +  5^  +  cx^-\- dx*  will 

bave  a  common  quadratic  factor  if 
(a  +  d)(a-dy=:(b-  c) (bd -  ac). 

6.  x^  -\-px^  -{-  qx-\-r  will  be  divisible  by  x'  -\-  ax-\-h  if 

a^  —  2pa'  +  {p^  -\-  q)  a  +  r — pq  =  0, 
and  b^  —  qb"^  +  rpb  —  r'^  =  0. 

7.  x^  -\-px  +  q  will  be  divisible  by  x'^  -\-ax  +  b  if 

a^-^qa'  -=^^  and  {b'  +  ^)  {b'  -  qf  -^p'b\ 

Determine  tbe  condition  necessary  in  order  tbat : 

8.  x^-{-4:px^-]-6qx'^-\-4:rx-{-t  may  bave  a  square  factor. 

9.  ax^-\-A. bx^ -\-6cx'  +  4Ldx+e  may  bave  a  complete  cube 

as  factor. 

10.    x^  +  10^^^  +  10 cx^  +  b  dx  +  be  may  bave  a  complete 
cube  as  factor. 


CHAPTER  III. 

DiEECT  Application  of  the  Fundamental  Formulas. 

p-     Formulas  [1]  and  [2].     (r^:d=3/)^==a;^=b2rry+3/^  etc. 

§  12.  From  this  it  appears  that  a  trinomial  of  which  the 
extremes  are  squares,  is  itself  a  square  if  four  times  the 
product  of  the  extremes  is  equal  to  the  square  of  the  mean, 
and  that,  to  factor  such  a  trinomial,  we  have  simply  to  con- 
nect the  square  root  of  each  of  the  squares  by  the  sign  of 
_the  other  term,  and  write  the  result  twice  as  a  factor. 


Examples. 

1.  4:x'  -  80:r^3/^  +  400y*  =  (2x'  -  SOy^  (2x'  -  20y^). 

2.  1  -  I2x''7f  +  36 ^V*  =  (1  -  6:r^?/^)(l  -Grr^y^). 

3.  {a-hy+{h-cf+2{a-h){h-c). 
This  equals 

{a  —  h  +b  —  c){a  —  h  -\-  h  —  c)  =  {a  —  c)(a  —  c). 

4.  x'^ -^-y"^ -{-z^ -\-2xy  —  2xz  —  2yz. 

Here  the  three  squares  and  the  three  double  products 
suggest  that  the  expression  is  the  square  of  a  linear 
trinomial  in  x,  y,  z. 

An  inspection  of  the  signs  of  the  double  products  ena- 
bles us  to  determine  the  signs  which  are  to  connect 
X,  y,  z;  we  see  that 

1.  The  signs  of  x  and  y  must  be  alike ; 

2.  The  signs  of  x  and  z  must  be  different ; 

3.  The  signs  of  3/  and  z  must  be  different. 

Hence,  we  have  x-\-y—z^  or  —x—y-\-z  =  —  (x-\-y  —  z)j 
and  the  factors  are  (x  -\-  y  —  z)  (x  -{-  y  —  z). 


78  FACTORING. 


Ex.  25. 

1.    9m'  +  12m  +  4;  c'"*-2c"^  +  l. 

3.    9a'b''  +  12abc  +  4:c'']  36^'y'- 24^/  +  4/. 

6.  ,^  +  (cc-yy-2z(x-y)-   (fJV  (^^j"- 2. 

7.  (:r^  -  yf  +  2(^^  -  3/)  (3/  -  .^)  +  (y  -  .^)l 

8.  (x'^  —  xyY  —  2(x^  —  xy)  (xy  —  y'^)  +  (^y  —  3/^)^. 

9.  (a  +  ^  +  ^)'-2c(a+Z>  +  (?)+c^  ^^p^-2p^q^ +  ^f 

10.  (3:r  -  4y)'  +  (2:r  -  83/)^  —  2(3:?;  -  4y) (2:?;  -  3y). 

11.  (:,^_,,3/  +  3/7+(a;^  +  :r3/  +  y7  +  2(r.^  +  a;V^+y0. 

12.  {bx'  +  2xy+1yy  +  {^x''+^yy 

-  2(4a;^  +  6y^)  {bx^  +  2xy  +  7y^). 


1m        f'lXl.n  /^\m—n 


14.  a2  +  Z>^  +  c'-2a5-2Z>c  +  2a(?. 

15.  a^  +  ^*  +  c'-2a^^2-2a'c^  +  2Z^2cl 

16.  {a-by  +  (b-  cY  +  {c-af  +  2{a-  b) (b  -  c) 

-2{a-b){c  -  a)  +  2{b  -c){a-  c). 

17.  4a'-12a'Z)  +  9Z^'  +  16a'c+16c'-245(?. 

Formula  [4].    x^  —  y'^  =  {x  +  y)  {x  —  y). 

§  13.  In  this  case  we  have  merely  to  take  the  square  root 
of  each  of  the  squares,  and  connect  the  results  with  the 
sign  +  for  one  of  the  factors,  and  with  the  sign  —  for  the 
other. 


FACTORING.  79 


Examples. 

1.  {a  +  hY-{c  +  d)\ 

This  =  [(a  +  h)  +  {c  +  d)]  [{a  +  h)-{c  +  d)] 
-=la  +  h  +  c  +  d){a  +  b  —  c  —  d), 

2.  Factor   (x^  -{-bxy-{-  y^y  —  {x'  —  xy  ■\-  y'^f. 

Here  we  have  \{x^  -{- b  xy  -\-  y^)  +  (x^  —  xy  +  3/^)] 
X  [(2?'  +  5^y  +  y2)  —  (>  —  :ry  +  ?/')] 
^^ix"  +  xy  +  y'')(^Qxy) 
=  l2xy{x  +  y)\ 

This  =  o?  —  {b  —  cf^  {a  +  b  -  c){a~b  +  6). 

4.    Resolve    (a^  +  ^>^/ -  (a^  -  ^^7  -  (a^  +  ^^  -  c^. 
This=:4a^^^2-(a2  +  ^2-c7 

=:  (2a&  +  a^  +  ^'  -  ^')  (2a5  -  a^  -  Z>^  +  e^). 
The  former  of  these  factors 

=  {a  +  by-(^  =  {a+b  +  c){a  +  b-c)', 
and  the  latter 

=^  c" -  {a-by  =  {c  +  a  -  b){c -  a  +  b). 
.*.  the  given  expression 

=  {a+b  +  c)(a  +  b  —  c){c  +  a-b){G  —  a  +  b). 


Ex.  26. 

9.  81a*- 1. 

10.  a'~lU\ 

11.  a}^-¥\ 

12.  a'-^>2+2^>(?-cl 

13.  (a +  2^)2- (3:^ -4y)^ 

14.  {x'  +  yy-4Lx'y\ 

15.  (:r  +  y)'-42^ 

16.  {^x  +  by-{bx  +  2>)\ 


1. 

49^2  _  4^2^ 

2. 

9a'-\b\ 

3. 

81a*-165^ 

4. 

100:^^-36yl 

5. 

5a^5-205a;^yl 

6. 

9a;^-16y^ 

7. 

^c'-l. 

8. 

^y'-ix'z\ 

80  FACTOEING. 


I 


17.  \x'y'-{x'-^f  —  z'')\ 

18.  {p(?  -\-  xy  —  y^y  —  {pi?  —  xy  —  y'^y, 

19.  {x''  —  y''  +  zy  —  ^x^z\ 

20.  {a  +  h  +  c  +  dy~{a-h  +  c-d)\ 

21.  {2+'^x  +  ^xy  —  {2  —  ?>x  +  4:xy. 

22 .  {p?+y  +  ^  ahy  -  (a'  +  h^)\ 

23.  (a^  -  ^2  +  c^  -  d'^y  -(2ac-2 hdy. 

24.  (a;2-3/2-27-4y'2l 

25 .  (a«  -  a^  Z)^  +  bj  -  (a«  -  5  a^  ^)^  +  bj. 

26.  a^'  -  Z^^'  +  6a'b'  -6b^a'  +  8b^a'-SaFb\ 

27.  (x'^  -\- y'^  + '^  —  ^y  —  y^  —  ^^y  —  ip^y  +  yz  +  zxy, 

28.  (x''  +  y^  +  z^  —  2xy  +  2xz  —  2yz)  —  (y  +  zy, 

29.  2a'Z>'  +  2Z>'c'  +  2c'a'-a*  — 5^-c\ 

30.  :z;*  +  2/*  +  2*  — 2a;'3/'  — 2y'2'  — 22'ar^. 

Formula  [A].     (a7+r)(:r+s)  =  x^-^(r^s)x-\-rs. 

Examples. 

1.  ^'«-9a;H20-(:r^-5)(r^^  +  4). 

2.  {x-yy-^x-y-\\^-=^{x-y-^\\){x-y~\^). 

3.  (a^ -  a^)  +  ^^7  +  6^)(a'  -  a5  +  5^)  - 4a^  +  9^>'^ 

-  [(a^  -  a5  +  b^)  +  (2a  +  3Z))] 
X  [(a'  -  a5  +  ^>^)  -  (2a  -  3^)]. 

4.  (ar^-5^y-6(:?;^-5a;)-40 

=  (a;^  -  5a;+ 4) (^  -  5^  -  10). 

5.  {ax  -\-by-\-  cy  —  {^n  —  n)  {ax  -\-by  +  c)  —  mn 

=  {ax  +  by  +  c  —  m)  {ax  +  by  +  c  +  n). 


i 


FACTORING.  81 


§  14.  It  will  be  seen  that  the  first  (or  coiniiiori)  term  of 
the  required  factors  is  obtained  by  extracting  the  square 
root  of  the  first  term  of  the  given  expression,  and  that  the 
other  terms  are  determined  by  observing  two  conditions  : 

I.  Their  product  must  equal  the  third  term  of  the  given 
expression. 

II.  Their  algebraic  sum  multiplied  into  the  common 
term  already  found  must  equal  the  middle  term  of  the 
given  expression. 

Hence,  to  make  a  systematic  search  for  integral  factors 
of  an  expression  of  the  form  x^:^hx±c,  we  may  proceed  as 
follows : 

1.  Write  down  every  pair  of  factors  whose  product  is  c. 

2.  If  the  sign  before  c  is  +,  select  the  pair  of  factors 
whose  suTu  is  b,  and  write  both  factors  x^,  if  the  sign  before 
^  is  +  ;  ^— ,  if  the  sign  before  b  is  — . 

3.  But  if  the  sign  before  c  is  — ,  select  the  pair  of  factors 
whose  difference  is  b,  and  write  before  the  larger  factor  x-\- 
or  X—,  and  before  the  other  factor  x—  or  rr+,  according 
as  the  sign  before  b  is  -j-  or  — . 

Examples. 

1.  x'^-\-^x-\-20.     The  factors  of  20  in  pairs  are  1  and  20, 

2  and  10,  4  and  5.  The  sign  before  20  is  +;  hence, 
select  the  factors  whose  swtn  is  9.  These  are  4  and 
5.  The  sign  before  9  is  +;  hence,  the  required  fac- 
tors are  (r^+4)(a;  +  5). 

2.  x^  —  d>x-\~12.     Pairs  of  factors  of  12  are  1  and  12, 

2  and  6,  3  and  4.  Sign  before  12  is  + ;  therefore 
take  the  pair  whose  sum  is  8.  These  are  2  and  6. 
Sign  before  8  is  — ;  hence,  the  factors  are  (x  —  2) 


82  FACTORING. 


3.  ^i''^- 21a; -100.     Pairs  of  factors  of  100  are  1  and  100, 

2  and  50,  4  and  25,  5  and  20,  10  and  10.  Sign  be- 
fore 100  is  — ;  therefore,  take  the  pair  whose  differ- 
ence is  21.  these  are  4  and  25.  The  sign  before  21 
is  —  ;  therefore,  x—  goes  before  25,  the  larger  factor, 
and  the  factors  are  (a;+4)(a;— 25). 

4.  x"  +  12a;  -  108.     Pairs  of  factors  of  108  are  1  and  108, 

2  and  54,  3  and  36,  4  and  27,  6  and  18,  9  and  12. 
Sign  before  108  is  — ;  therefore,  take  the  pair  whose 
difference  is  12.  These  are  6  and  18.  Sign  before 
12  is  + ;  therefore,  x-\-  goes  before  18,  the  larger 
factor,  and  x—  before  6,  the  other  factor.  Hence, 
the  factors  are  (a;—  6)  (a; -f  18). 

Note.  It  will  be  found  convenient  to  write  the  factors  in  two 
columns,  separated  by  a  short  space.  Taking  Exam.  2  above,  pro- 
ceed thus  :  Since  the  sign  of  the  third  term  is  +,  write  the  sign  of 
the  second  term  (in  this  case  — )  above  both  columns. 

1  12 

(x  -  2)  {x  -  6) 

Exam.  3  above.  Since  the  sign  of  the  third  term  is  — ,  write  the 
sign  of  the  second  term  (in  this  case  — )  above  the  column  of  larger 
factors,  and  the  other  sign  of  the  pair,  ±,  above  the  other  column. 

+  - 

1  100 

2  50 
(a; +  4)         {x-2b) 

5.  a;' -34a; +  64. 

Here  we  have  the  factors 

1  64 

a;  -  2         a;  -  32 
4  16 

And  since  the  last  term  has  the  sign  +,  and  the 
middle  term  has  the  sign  — ,  we  write  —  over  both 
columns. 


FACTORING.  83 


6.  x'+l2x-Q4:. 

+ 

1  64 

2  32 
X  —  4       ^+16 

Here,  since  the  last  term  has  the  sign  — ,  we  write  the 
sign  (+)  of  the  middle  term  over  the  column  of 
larger  factors,  and  the  sign  —  over  the  other  column. 

7.  ^*-10:x;'-144. 

Here  we  have  the  pairs  of  factors 

+ 

1  144 

2  72 
4  36 

^  +  8  ^^  -  18 

And  since  the  sign  of  the  third  term  is  — ,  we  write 
the  sign  of  the  second  term  (in  this  case  — )  above 
the  column  of  larger  factors,  and  the  other  sign  (of 
the  pair,  ±)  above  the  other  column. 

Ex.  27. 

1.  x^-bx—\^\  ^2  -  9;r  +  14  ;  x^-^-lx-^  12. 

2.  :r'-8:?;  +  15;  ri;'— 19a;  +  84;  ^'-7;r-60. 

3.  4^'-2:r-20;  9:^^-150:^  +  600. 

4.  \x^-\-\\x-m',  25:^2 +  40:?; +15;  9  :^«  -  27  :z;H  20. 

5.  ^x^-\-\\x^Vl\  16  a;* -42^' -20. 

6.  x^-(d-\-})')x^-\-a^h'",  ^{x-\-yf-^(x-^y)-m. 

7.  {x^  +  fj  -  {(f  -  h"-)  {x"  +  %f)  -  o}  h\ 

8.  {a  +  hy~2c{a  +  h)-?>c\ 

9.  (:.  +  yy  +  2{x'  +  f)  {x  +  y)  +  {x^  -  yj. 


84  FACTORING. 


10.  {a  +  hy-^ah{a  +  b)  -  {p}  -  hy. 

11.  {x^  +  xy  +  y^  +  ^"^  —  y^  —  ^^y  ~  2y^  —  2r?;^ 

12.  a'  -2a(h  —  c)-S(b—  c)\ 

13.  (a;'  +  y7  +  2a'(:^^  +  /)  +  «'-^'. 

14.  (.^^-10.r)^-4(ri;'^-10:i;)-96. 

15.  (:r^  -  14:?;  +  40)^  -  25  (^-^  -  Ux  +  40)  -  150. 

16.  (x'  —  xy  +  7/y  +  2xy(x''  —  ri^y  +  /)  -  Sx'2j\ 

17.  2^- 3^' +  2;  r?;*-2:r'-3;  9x' +  9x'y' -~  107/\ 

18.  c'"^  +  c"^  -  2  ;  ^«  -  a;'  —  2  ;  x''''  -  2x'^y''  -  Sy''\ 

19.  ^^'"'  —  (a  —  b)  x'^y""  —  aZ?/^\ 

§  15.  Trinomials  of  the  form  ax"^ -J- bx -{- c  (a  not  a  square) 
may  sometimes  be  easily  factored  from  the  following  con- 
siderations : 

The  product  of  two  binomials  consists  of 

1.  The  product  of  the  first  terms  ; 

2.  The  product  of  the  second  terms  ; 

3.  The  algeh^aic  sum  of  the  products  of  the  terms  taken 
diagonally. 

These  three  conditions  guide  us  in  the  converse  process 
of  resolving  a  trinomial  into  its  binomial  factors. 

Examples. 

1 .    Eesolve  6  x''  —  13  :ry  +  6  tj\ 

Here  the  factors  of  the  first  term  are  x  and  6^,  or  2x 
and  3:?;;  those  of  the  third  term  are  y  and  6?/,  or 
2y  and  3?/.    These  pairs  of  factors  may  be  arranged : 
(1)  (2)  (3)  (4) 

x  2x  y  2y 

^x  ^x  6y  3y 


FACTORING.  85 


Now,  we  may  take  (1)  with  (3)  or  (4),  or  (2)  with  (3) 
or  (4)  ;  but  none  of  these  combinations  will  satisfy 
the  third  condition.  If,  however,  in  (4)  we  inter- 
change the  coefficients  2  and  3,  then  (2)  and  (4)  give 

2x     3y 
and  ?>x      2y, 

where  we  can  combine  the  "  diagonal  "  products  to 
make  13,  and  the  factors  are 

and  '^x  —  2y. 

The  coefficients  of  (2),  instead  of  those  of  (4),  might 
have  been  interchanged,  giving  the  same  result. 

2.  6^^~15^y  +  63/^ 

Here,  comparing  (2)  and  (3),  Exam.  1,  we  see  that 
their  diagonal  products  may  be  combined  to  give 
15,  and  the  factors  are  2x  —  y  and  3  a;  —  63/. 

3.  6a;'  — 20a;y  +  6yl 

Here,  again  referring  to  Exam.  1,  we  see  at  once  that 
it  is  useless  to  try  both  (2)  and  (4),  since  the  diag- 
onal products  cannot  be  combined  in  any  way  to 
give  a  higher  result  than  l?>xy.  But  comparing  (1) 
and  (4),  we  obtain,  by  interchanging  the  coefficients 
in  4,  ^_^y 

and  6a;— 23/, 

which  satisfy  the  third  condition. 

Or,  we  might  interchange  the  coefficients  of  (3),  and 
take  the  resulting  terms  with  (2),  getting 

2a;  —  6y 
and  3  a;—    y. 


86  FACTOEING. 


Here  the  large  coefficient  of  tlie  middle  term  shows  at 
once  that  we  must  take  (1)  and  (3)  together.  Inter- 
changing the  coefficients  of  (1),  we  have 

^x—    y 
and  x-\~Qy. 

The  same  result  will  be  obtained  by  interchanging  the 
coefficients  of  (3). 


Qx^  —  l^xy  ~  6y", 
6x'^-{~bxy  —  6y'^. 
b6x''  +  562xy  +  20y\ 
b6x''  —  122xy  +  20y\ 
56x''  —  102xy  —  20y\ 
b6x''-229xy  +  20y\ 
56^'-94:ry  +  20yl 

18.    56x'  —  276xy-207/. 
S6x'-S3xy-157/. 

20.    72x^~19xy-A0y\ 

§  16.  Generally,  trinomials  of  the  form  ax^  +  bx-^-c  (a  not 
a  square)  may  be  resolved  by  Formula  [A] ;  thus, 

Multiplying  by  a  we  get  a^x^  +  hax  +  ac.  Writing  z  for 
ax,  this  becomes  z^-{'hz-\-  ac.  Factor  this  trinomial,  restore 
the  value  of  z,  and  divide  the  result  by  a. 

Examples. 
1.    6:^-'  + 5^  —  4. 

Multiplying  by  6,  we  get  (6:r)2+5(6^)-24  or  z''+5z-24:. 

Factoring,  we  get  (z  —  8)  (2;  +  8) ;  hence,  the  required 

factors  are  |(6a;-  3) (6x  +  S)  =  (2^-1) (3^  +  4). 


Ex. 

28. 

1. 

6x'-S7xy  +  67f. 

11. 

2. 

6af  +  9xy-6y\ 

12. 

3. 

56x'-16xy  +  20y\ 

13. 

4. 

b6x''-d6xy-20y\ 

14. 

5. 

56a;^-1121a:y  + 203/1 

15. 

6. 

b6x'-6Sxy  +  20y\ 

16. 

7. 

56x'-b5Sxy--20y\ 

17. 

8. 

66x''  +  Se>xy-20y\ 

18. 

9. 

b6x'-67xy  +  20y\ 

19. 

0. 

56x'  +  Sxy-20y\ 

20. 

FACTORING.  87 


Factoring  2;^  —  13  2;?/  +  36  y^,  we  get  (2;  —  4y)  (2;  —  9y) ; 
hence,  the  required  factors  are 
i(6^-4y)(6x-93/)-(3a;-2y)(2a;-32/). 

3.    33  — 14:^-40 rrl 

Factoring  1320  -  14^  —  z\  we  get  (30  -  2;)  (44  +  z) ; 
hence,  the  required  factors  are 
^V(30- 40^)  (44  + 40a;) -(3 -4a;)  (11 +  10^). 

Note.  The  factors  may  conveniently  be  arranged  in  two  columns, 
each  with  its  appropriate  sign  above  it. 

-  + 

Exam.  1,  above :  1  24 

■     2  12 

J(6a;  -  3)(6a;  +  8)  =  {2x  -  l)(3a;  +  4). 

Exam.  2,  above :  1  36 

2  18 

3  12 

•    J(6a;-4)(6a;-9)  =  (3a;-2)(2aj-3). 

Another  method  of  factoring  trinomials  of  the  form  ax^  -\-hx  -\-  c 
is  as  follows : 

Multiply  by  4  a,  thus  obtaining  4  a''^a;2  +  4a5a;  +  ^ac.  Add  h^  —  b^, 
which  will  not  change  the  value,  4a^a;^  +  4:abx  +  6^  —  6^^  +  4ac ;  by 
[1]  this  may  be  written  (2  ax  +  bf  —  (b'^  —  4ac).  Factor  this  by  [4], 
and  divide  the  result  by  4  a. 

Example.    Factor  56  x"^  +  137  a;-  27,885.     Multiply  by  4  x  56  or 
^2  X  112,  1122 a;2  +  2  X  137  X  112a;  -  6,246,240.  Add  137^  - 137^  then, 
1122  a;2  +  2  X  137  X  112  a;  +  137^  -  (137^  +  6,246,240) 
=  (112a;  +  137)2-6,265,009 
=  [(112a;  +  137)  +  2503]  [(112a;  +  137)  -  2503] 
=  (112a;  +  2640)(112a;  -  2366). 

We  multiplied  by  4  X  56 ;  we  must,  therefore,  now  divide  by  that 
number.     Doing  so,  we  obtain  as  factors  (7  a;  +  165)  (8  a;  — 169), 


FACTORING. 


Ex.  29. 

1.  10a;^  +  rr-21.  8.    15a' +lSa'b' -20b\ 

2.  10^*'-29:z;-21.  9.    12^2-^-1. 

3.  10^'  +  29:r-21.  10.    O^r^y' —  3rry'  — 6y*. 

4.  6:r'  — 37^  +  55.  11.    40;^  + 8;2;3/  + 3yl 

5.  12a' -5a -2.  12.    6 5^ :r^  -  7 ^^^  -  3 rz;^ 

6.  12^^  — 37;r+21.  13.    6x' -  x'f -S5y\ 

7.  12rr2  +  37a;  +  21.  14.   2^* +  ^^-45. 

15.  4:x'-S1x''f  +  9i/\ 

16.  4(^  +  2/-37^(^  +  27  +  9r?;^ 

17.  6(2x  +  3yy  +  5(6;r^+  5:ry  -  6f)  -6(Sx~  2y)\ 

18.  6(2:?;  +  Zyf  +  5 (6:?;'  +  5xy  —  ^y^J -^{?>x-  2y)\ 

19.  ^{x''+xy+yy+l?>{x'+x^y''+y')-^^5{x'-xy+y'')\ 

20.  21  {x"  +  2xy+  2^^)'-  6 (a;'-  2a:y  +  2/)'-  5  {x^+4:i/). 

Extended  Application  of  the  Formulas. 

§  17.  The  methods  of  factoring  just  explained  may  be 
applied  to  find  the  rational  factors,  where  such  exist,  of 
quadratic  multinomials. 

Examples. 
1.    Kesolve  12:?;2  —  xy  —  20^/'  +  8^  +  41y  -  20. 

In  the  first  place  we  find  the  factors  of  the  first  three 
terms,  which  are 

^x-\-  5y 
and  ?>x  —  4:y. 

Now,  to  find  the  remaining  terms  of  the  required  fac- 
tors, we  must  observe  the  following  conditions : 


FACTORING.  89 


1.  Their  product  must  =  —  20. 

2.  The  algebraic  sum  of  the  products,  obtained  by 
multiplying  them  diagonally  into  the  y's,  must  =  il  y. 

3.  The  sum  of  the  products,  obtained  by  multiplying 
them  diagonally  into  the  x^,  must  =  ^x. 

We  see  at  once,  that  —4,  with  the  first  pair  already 
found,  and  +  5,  with  the  second  pair,  satisfy  the 
required  conditions  ;   and  hence  the  factors  are 

4rr+53/ —  4 
and  307  — 4?/ +  5. 

.    p'  +  2pr-2q^  +  lqr-?>r''+pq. 

Here  the  factors  of^^  -\-P9.  —  2^^  are 
p  +  2q 
and  P~  ^^ 

Now  find  two  factors  which  will  give  —  3  r^,  and  which, 
multiplied  diagonally  into  the  ^'s  and  ^-'s  respec- 
tively, will  give  2pr  and  7  qr ;  these  are  found  to  be 
—  r,  taken  with  the  first  pair,  and  +  3  r,  taken  with 
the  second  pair.     Hence,  the  required  factors  are 

p-\-2q  —  T 
and  p  —  q  +  ^r. 

The  work  of  seeking  for  the  factors  may  be  conveniently 
arranged  as  follows : 

3.    x^  +  xy~'2.y''  +  2xz  +  lyz-^z\ 

Keject : 

1.  The  terms  involving  z  ; 

2.  The  terms  involving  y  ; 

3.  The  terms  involving  x ; 

and  factor  the  expression  that  remains  in  each  case. 


90  FACTORING. 


1.  x^^xy-'2.y'^(x  —  y){x-\-'2.y). 

2.  x^-\-1xz-'^z^  =  {x  +  Zz){x-z). 

*3.   -2y^  +  73/z-3.^=(--y  +  3^)(23/-.). 

Arrange  these  three  pairs  of  factors  in  two  sets  of  three 
factors  each,  by  so  selecting  one  factor  from  each 
pair  that  two  of  each  set  of  three  may  have  the  same 
coefficient  of  x,  two  may  have  the  same  coefficient  of 
2/,  and  two  the  same  coefficient  of  z  (coefficieyit  includ- 
ing sign).     In  this  example  there  are 

x~y,       x  +  ?>z,     -y  +  ^z, 
and  x-\-2y,     x—z,         2y  —  z. 

From  the  first  set  select  the  common  terms  (including 
signs)  and  form  therewith  a  trinomial,  x  —  y -^-Sz. 

Eepeat  with  the  second  set,  and  we  get  x-{-2y  —  z. 

.'.  x'^  +  xy-  2y''+2xz  +  lyz  -  Sz' 
^(x-y  +  Sz)(x  +  2y~zy 

4.  3x''~8xy-~3y''  +  S0x  +  27, 

1.  Sx''-8xy~Sy''  =  (Sx  +  y)(x~dy). 

2.  Sx''  +  30x  +  27  =  (3x  +  ^)(x  +  9). 

3.  -Sf  +  27  =  (y  +  3)(-Sy  +  9). 

.'.  the  factors  are  (3^  +  ?/  +  3)  (r?;  —  3y  +  9). 

5.  6a'-7ah  +  2ac~20b'  +  QUc-A8c\ 

1.  6a^-7ab~20b''=:(2a~bh)(Sa  +  4:h). 

2.  6a'  +  2ac--^Sc'  =  (2a  +  6c)(Sa-Sc). 

3.  -20h'  +  e>4:hc-4:Sc'  =  (-5b+6c)(4:h-8c). 
.-.  the  factors  are  {2a-5b  +  6c)(Sa  +  4cb~-8c). 

To  find,  where  such  exist,  the  factors  of 

ax"^  +  bxy  +  cxz  -\-  ey^  +  gyz  +  hz^. 
Multiply  by  4  a  : 

4  a^  :r^  +  4  abxy  +  4  acxz  +  4  aey"^  +  4  agijz  +  4  ahz^. 


FACTORING.  91 


Select  the  terms  containing  x,  and  complete  the  square : 
thus, 

4a^;r^  +  4:abx7/  +  ^acxz  +  b'^if  +  2bcxz  +  c'^z'^ 

-  (b'  -  ^ae)f  -2(hc-  2ag)yz  -  (c^  -4:ah)z^ 
=  {2ax  +  by  +  czy 

-  [(^^  -  4ae)y'  +  2{hc-  2ag)yz  +  {c"  -  ^ah)z^\ 

If  the  part  within  the  double  bracket  is  a  square,  say 
{r)iy  +  nzy,  the  given  expression  can  be  written 

(2  ax  -\-  by  -{■  czy  —  {my  +  nzy, 

which  can  be  factored  by  [4].  Factor  and  divide  the  result 
by  4  a.  If  the  part  within  the  double  bracket  is  not  a 
square,  the  given  expression  cannot  be  factored.  If  h  and  c 
are  both  even,  multiply  by  a  instead  of  by  4  a,  and  the  square 
can  be  completed  without  introducing  fractions.  If  e  be 
less  than  a,  it  will  be  easier  to  multiply  by  4e  instead  of 
by  4  a,  and  select  the  terms  containing  y.  A  similar  remark 
applies  to  h. 

This   method   can   evidently  be  extended  to   quadratic 
multinomials  of  any  number  of  terms. 


I] 

I 


Examples. 

1.    Eesolve  x^  -\-  xy  '\-  2xz  —  ^y"^  -\-  lyz  —  ^z^  into  factors. 
Multiply  by  4  : 

4a;2  +  4.xy  +  ^xz  -  ^f  +  2^yz  -  12^1 
Complete  the  square,  selecting  terms  in  x  : 

A:x'  +  4,xy  +  Sxz  +  y''  +  ^yz  +  4:z^--'^y''  +  2^yz-l^z^ 

=  {2x  +  y  +  2zy  -  (3y  -  4^)^ 

=  [{2x+y+2z)  +  {^y~^z)][(2x+y+2z)-{2>y-^z)-\ 

-=  {2x+^y-2z){2x-2y+Qz)  -  ^{x+2y-z){x~y+Zz). 
.'.  the  factors  are  {x  -{-  2y  —  z)  {x  —  y  -{■  ^ z). 


92  FACTORING. 


2.  6a'~7ah  +  2ac-20b'  +  64:bc-4:Sc\ 
Multiply  by  4  X  6  =  24  : 

144  a^  -  168  aS  +  48a^  -  480  ^^  +  15365(7  -  1152  c^ 
=  (12a  -  lb  +  2cy  -  529 5^  +  IbQUc -  llbQc' 
=  ll2a-7b  +  2cy-(23b-d4:cy 
=  (12a+  165  -  32c)(12a  -  305  +  36c) 
=  24(3a  +  45-8c)(2a-55  +  6c). 
.*.  the  factors  are  3a  +  45  —  8c  and  2a  —  55  +  6^?. 

3.  x''  +  12xi/  +  2xz  +  262f  —  S7/z-9z^ 

=--  (x'+  12xy+2xz+36y'+  i2yz+z')-10f-207/z-l 0/ 
=  (x  +  e^  +  zy-[(y  +  z)^-10f 
=  [x+(6+^10)y  +  (^10  +  l)z] 

x[:.+  (6-viO)y-(VlO-l)4 

4.  3a'  +  10r^5  -  Uac  +  12ac?-  85-  -  Sbd+Sc'  -8crl 
Multiply  by  3,  not  4x3,  since  the  coefficients  of  the 

other  terms  in  a  are  all  even  : 

9a'  +  30a5  -  42ac  +  36ac^  -  245'  -  245^+ 24^'-  24:cd. 

Select  the  terms  containing  a,  and  complete  the  square  : 
(3a  +  5b-7c  +  6dy 

-A9b'+70bc-S4:bd-25c'  +  60cd-36d' 
=  (3a  +  5b-7c  +  6dy-(7b-5c  +  6dy 
=  l3a+12b-12c  +  12d)(Sa-2b-2c) 
=  S(a  +  Ab~4:C  +  Ad)(3a-2b-2c). 

.'.  the  factors  are  a  +  45  — 4c  +  4(i  and  3a  — 25  — 2c. 

Ex.  30. 

1.  7a;'-:i;3/-6y'-6:r-20y-16. 

2.  20:r'-15:ry-53/'  — 68.^  —  422/ -88. 

3.  3:z;*  +  a;'y'-4y'  +  10a;'-17y'-13. 

4.  20^'-20?/'  +  9a:3/  +  28.r  +  35?/. 


FACTORING.  93 


5.  72^'-8y'  +  55^y+123/-169a;  +  20. 

6.  x^ —  xy —\2y'^ —  bx —  Iby. 

7.  ^x^+l^xy  +  ^y''  +  2xz~z\ 

8.  ^x^  +  ^y''-l?>xy-^z'-2yz  +  ^xz. 

10.  Ibx'  -  IGy*  -  22^'y''  +  Ibz'  +  \^fz^  +  mx'z\ 

11.  ^a^-\h¥  —  ^ab-'2X&~?>Uc-^ac. 

12.  a^  +  ^'  +  c'-2a^Z^^-2Z)^c^-2c^al 

§  18.    Trinomials  of  the  form  ax^  -\-hx^  -\-  c  can  always  be 
broken  up  into  real  factors. 

If  a  and  c  have  different  signs,  the  expression  may  be 
factored  by  §  16. 

If  a  and  c  are  of  the  same  sign,  three  cases  have  to  be 
considered  : 

(i.)  Z>-2V(«^)- 
(ii.)  Z)>2VM. 
(iii.)    h<2-^{ac). 

Case  I.     h  =  ^  ^{ac).     This  case  falls  under  §  12,  For- 
mula [1],  where  examples  will  be  found. 

Case  II.     Z>  >  2  -yj(ac).     This  case  falls  under  §  16,  where 
examples  will  be  found. 

The  following  additional  examples  are  resolved  by  the 
second  method  of  that  section  : 

Examples. 
1.    ^x'  +  bx^y''+y\ 

Here  we  see  that  {^y'^y  will  make,  with  the  first  two 
terms,  a  perfect  square,  and  we  therefore  add  to  the 
given  expression,  (f  y^)^  —  (|■3/^)^ 


94  FACTORING. 


The  expression  then  becomes 

=  (2x^  +  ^yJ-^y^ 

=  (2x'  +  2f)(2x^  +  ^f) 
=  {x'  +  y')(Ax'  +  f). 

2.  3x'+6x'  +  2. 

Here,  multiplying  by  4  X  3,  and  completing  the  square 
as  in  Exam.  1,  we  have 
36a;* +72^^62  + 24 -6^ 
-(6:^^  +  6)2-12 

=  l6x'  +  6~  V12)  (6^2  +  6  +  V12), 
which,  divided  by  4  X  3,  gives  the  required  factors. 

3.  ax*  +  hx'^  +  c. 

Proceeding  as  in  Exam.  2,  we  have,  by  multiplying 
by  4  a, 

ax^  -}-bx'^  -\-  c 

=  (4:a''x'  +  4:abx''+h^ -  5^  +  4a(?)  -^  4a 
^  [2ax''+b+'y/(b'-Aac)]  [2ax''+b-^(b'-4:ac)]  -^4a. 

Ex.  31. 

1.  x^  +  lx'  +  l]  ^x'  +  Ux'  +  l. 

2.  x*+7x''y''  +  y']  3x'  + 5x''y^ +  y\ 

3.  4:x'  +  lOx''  +  S]  Six  +  yy  +  bz'ix  +  yy  +  z', 

4.  x'+7x'^y'  +  S\y';  x'  +  7x'i/  +  S^7j\ 

5.  4:x'  +  9x^y'  +  {iy']  4:(a  +  by  +10c'(a  +  bf  +  3c\ 

6.  Sx'  +  Sx'^y'+^-^y']  36:r*  +  96r^ +  55. 

7.  5:r*  +  20:c'^  +  2;  4a*+12a2  +  l. 

8.  4:(x  +  7/y+12(x  +  yyz'  +  z']  5x' +  20x'y' +  2y\ 


FACTORING.  95 


9.  9x'+Ux'  +  4t;  2x'  +  12x'(7/  +  zf  +  lb(y  +  z)\ 

10.  2:r*+12r?;^  +  15;  7x' +  ^0x'' +  4:5. 

11.  8:r*  +  36r?;'3/'  +  293/^  7^*  +  20:?;'3/'- 20?/*. 

12.  7(a-^)*  +  16(a-^7c^+5c^  fa*  +  3a^^>^  +  ^^ 

13.  Sx'  +  6x'f  +  2y'',  S(ia  +  by  +  6(a^~bJ  +  2(a-by. 

14.  49a*-84a^Z^^  +  22&^  25m*  + 60m^7i^  +  27?i\ 

,  15.  49(m  +  ny-84(m2-n7  +  22(m-7z)*. 

i 

^      Case   III.     b  <  2-y/(ae).      This   case   may   be   brought 

under  §  13. 

The  following  examples  illustrate  the  process  of  reduc- 
tion and  resolution  : 

1.  a:^-7a^  +  l.  Examples. 

^  We  have  to  throw  this  into  the  form  d^  —  b'^: 

I   '  x'-7x''+l  =  (x'  +  iy-9x^ 

=-  (^^  +  1  +  Sx)  (x'  +  l-  Sx). 

2.  9x'  +  Sx'f  +  4y*  =  (Sx'  +  2y7  -  9x'y'^ 

=  (3:r'  +  2y'  -  dxy)(Sx'  +  2y'+3xy). 

3.  :t^*  +  3/*  =  (:r^  +  y7~2a;^3/^ 

.^  (^2  _^  ^2  _,_  ^^  y2)  (x^  +  y^  -  ^3/ V2). 

==  (^'  +  y'  +  f  ^y)  (^'  +  3/'  -  f  ^y). 
5.    ax'  +  bx'  +  c=  (x'^a  +  V^)'- (2 V^- ^) x" 
=  lx'^a+-^c~C<2V^c-b)xl 

§  19.  It  is  seen  from  these  examples  that  we  have  merely 
to  add  to  the  given  expression  what  will  make  with  the 
first  and  last  terms  (arranged  as  in  Exam.  5)  a  perfect 
square,  and  to  subtract  the  same  quantity.     In  Exam.  2, 


I 


96  FACTORING. 


for  instance,  the  sq  uare  root  of  9x^  =^3  x^,  the  square  root 
of  4 3/^^=2 y'^;  hence,  ^x^-\-2y'^  is  the  binomial  whose  square 
is  required;  we  need,  therefore,  12x'^y^]  but  the  expression 
contains  2>x'^y'^\  hence,  we  have  to  add  and  subtract  12x^y'^ 
—  ?>x''y^=9x^y\ 

Hence,  we  derive  a  practical  rule  for  factoring  such 
expressions : 

1.  Take  the  square  roots  of  the  two  extreme  terms,  and 
connect  them  by  the  proper  sign ;  this  gives  the  first  two 
terms  of  the  required  factors. 

2.  Subtract  the  middle  term  of  the  given  expression  from 
twice  the  product  of  these  two  roots,  and  the  square  roots 
of  the  difference  will  be  the  third  terms  of  the  required 
factors. 

6.  x'  +  ^x''f  +  y\ 

Here  -y/:r*  =  x^,  y/y^  ~  y^  and  the  first  two  terms  of 
the  required  factors  are  x^  +  y'^ ;  twice  the  product 
of  these  is  +2ri;^y^,  from  which,  subtracting  the  mid- 
dle term,  -j^^^y^,  we  get -f-|-:r^3/^ ;  the  square  roots 
of  this  are  ±  \xy.  Hence,  the  factors  are 
0?  -\-y'^  ^  \xy. 

Note  that,  since  ^y'^  =+3/^,  or  —  y^,  it  may  sometimes 
happen  that  while  the  former  sign  will  give  irrational 
factors,  the  latter  .will  give  rational  factors,  and  con- 
versely. 

7.  x^  —Wx^y'^  -\-y^. 

Here,  taking  +3/^,  we  have 

^^  +  3/^  +  ^y  V^^  ^^^  x^  -\-  y'^  —  xy  y'lS. 
But,  taking  —  y^,  we  have 

x^  —  y"^  +  ^ocy  and  x^  —  y'^  —  Z  xy. 
Sometimes  hoth  signs  will  give  rational  factors. 


FACTORING.  97 


8.  l^x'-llx^tf  +  y\ 

Here  we  have  (4^;^  +  y^  +  3  xy)  (4:r^  +  3/^  —  3  xy), 
and  also        (4  x^  —  y^  +  5  xy)  (4a;^  —  y^  -  -  5  xy). 

tEx.  32. 

1.  x'  +  2x^y''  +  9y']  x' ~  x'' y'' +  y' ;  x'  +  x^y''  +  y\ 

2.  x'  +  4:y']  IQx'  +  y'-xy')  \x'  +  y\ 

3.  .2;*+l;  ^'  +  9y^   1-12^^  +  16/. 

4.  :r*-7:r2  +  l;  :r*  +  9;  ^^r^  +  y'- 30:^1 

5.  y'-x'+llxHf',  ^^4/;  ^*  +  4^2  +  16. 

6.  ^x'  +  y'-^\x'y'']  x' +  y' -^x'^y'' )  4:x'  +  l. 

7.  ^'"^+64/"*;  ^*"*  +  43/'"*;  i^^  +  i^y*  — 5f  :r'?/\ 

8.  4^'-8;r2  +  l;  7^^' -  i^r*  -  36/ ;  x^  +  aUj\ 

10.  16a;*- 2507^  +  9;  4:r' -  16^^;^  + 4  ;  13:i;y  -  9:r*-4y^ 

11.  4^*-12f|a;'y2  +  9/;  x'  +  ^x'  +  ^b. 

12.  a*  +  5*  +  (a  +  ^y;  l  +  a*  +  (l  +  a)\ 

13.  {x  +  yy-1z\x  +  yy  +  z\ 

14.  (a  +  ^)*  +  7c'  (a  +  5)2  +  g\ 

15.  16a*  +  4(5-c)*-96i'(5-c)^ 

16.  4(a  +  5y  +  9(a--5)*-21(a2-52)^ 

17.  {x'  +  f-xyy-1{x'  +  yj  +  {x  +  y)\ 

18.  (a^  +  a5  +  hj  +  7  (a^  -  Z>^)*^  +  (a  -  ^>)*. 

19.  16a*  +  4a2  +  l;  r^*-41:r2+16. 

20.  (a2  +  iy  +  4(a2  +  l)2a2  +  16a^ 

I  {x+lf  +  ^^x'  -  ly  +  ^(x-l)\ 


FACTORING. 


§20.    We  can  apply  [4],  §  13,  to  factor  expressions  of  the 
form  ax^  +  ^^^  +  ^^^  —  '^'^<^-     This  may  be  written, 

a  (x^  —  r'^)  -{- bx  (x'^  -{- r)  =^  [a  (x^  —  r)  +  hx'\  {x^  +  r). 

Examples. 

1.  6:^'  +  4^^+12;r-54 

^^{x'-^)  +  4:x{x''  +  ?>) 
=  (^^  +  3)[6(:r2-3)  +  4:r] 
=  (^2  +  3)(6a;^  +  4:z;-18). 

2.  11^*+ lOa;^- 40^27 -176 

=  ll(^^-16)  +  10:^(:r^-4) 
-=(:r^-4)[ll(^^  +  4)  +  10:i;] 
=  (:r^-4)(ll^'^  +  10:r  +  44). 

3.  ^Ox'  +  ?>0x^  +  mx -im 

=  10(4:r*  - 16)  +  lbx{2x''  +  4) 
=  {2x^  +  4)  [10  (2a;^  -  4)  +  Ibx'] 
=:(2r^  +  4)(20a;^  +  15^-40). 

Note.   To  determine  r,  take  the  ratio  of  the  coefficient  of  x^  to  the 
coefficient  of  x. 

Ex.  33. 

Kesolve  into  factors : 

1.  x^  +  23(^  +  ^x-^.  6.    10^* +  6:2;'+ 30^ -360. 

2.  2:?;*  + 2a;' +  6^ -18.  7.    i.r*  + 20:?;'  + 4:?;- y^. 

3.  ri;*  +  3^'+12:2;-16.  8.    25:?;*- 40^'  + 8^- 1. 

4.  3:i;*  +  ^'-4rr-48.  9.    ?>1^x'-~^0x^+^^x-^Q, 

5.  5:r*  + 4:^' -12.^-45.         10.    63ri;*-39a;'+52a;-112. 

11.  d>\0x'  +  ^-^x^  +  ^x-2^. 

12.  2i2a;*- 33a;' -3:^-2. 


FACTORING.  99 


13.  \x'  +  i-^x'~^x~^\. 

14.  ^0x'~?>2a^y  +  e)4:xij  —  ^20y\ 

15.  24:x'-l2x^y  +  ?>0xf~imy\ 

16.  2x'  +  \x^y-^xf-bl2ij\ 

17.  ll:?;*+10:r'-12.'?;-15|l. 

18.  40 :t'^  +  30 r'  +  60^'  -  160. 

19.  I?>x'-l2x^y+12xf-^^d>y\ 

20.  ?>x'  +  dx^y+12xi/-4cd>t/. 

21.  bx^  +  4^^y  —  120;?/^  —  45?/^ 

22.  4:2.-*  — 14:r>  +  28a;/-16/. 

23.  a;*  +  80:?;V+16^y3-2Vy'. 

24.  2x^~x^y  +  ^x2/-12y\ 

§  21.    Formulas  [1]  and  [4]  may  sometimes  be  applied  to 
factor  expressions  of  the  form 

ax'^  +  hx^  +  cx^  +  rhx  +  r^  a. 
This  may  be  put  under  the  form 

a  (^'*  +  t")  +  hx  (x'  +  r)  +  ex' 

=  a(x'^  +  ry  +  ^:z;(^^  +  r)  +  {c--  2ar)x', 
which  can  sometimes  be  factored. 

Examples. 

1.    :t'*  +  6  r'^  + 27:?;' +  162:^+729. 
^4_f_  729  +  ^x{x'  +  27)  +27:r' 
=  {x"  +  27)'  +  ^xix"  +  27)  +  9^'  -  36:^' 
=  (V  +  27  +  3:r)'  -  36  a;',  which  gives  the  factors 
x''-Zx-\- 27  and  :?;'  +  9^  +  27. 


100  FACTORING. 


'=  (x'  +  5/  +  4:x(x'  +  5)  -  6x' 
=  (x'  +  5)^  +  4:x(x'  +  5)  +  4:X'  ~  10^^ 
=  lx'  +  5  +  2x-  ^Vl(^)  (^''  +  5  +  2^  +  ^z^VlO). 

Ex.  34. 

Kesolve  into  factors  : 

1.  x'-6x^  +  27x^'  —  162x+729. 

2.  x'  +  2x'  +  ^x''  +  8x+16. 

3.  ^*  +  ;r'  +  ^'  +  ^-f  1. 

4.  :z;*  — 4:?;^  +  ^?;^  — 4:?;+ 1. 

5.  4^*-12^^-6^^-12:r  +  4. 

6.  x'+Ux'-25x'-70x  +  25. 

7.  16:r*  — 24:r^-16:r'+12:r  +  4. 

8.  a;*  +  5^'-16;r'  +  20:?;+16. 

9.  ^*  +  6:?;^-ll:z;'-12:?;  +  4. 

10.  ^*  +  4^'y  +  ^'y'  +  12:r/  +  93/*. 

11.  x^  +  6x'-9x'"  —  6x  +  l. 

12.  r^*  +  4:z;^?/  — 19:?;^2/'  +  4rry^  +  y*. 

13.  4:r*  +  4:?;V-65^'^3/'-10:r/  +  25y*. 

14.  :z;*  +  6:?;'y-9:r'y'--6:r/  +  3/\ 

15.  rr*  +  6^V+10:^V'  + 12^3/3  +  4/. 

16.  9x'  +  18x''7j-b2x''7/-12xf  +  A7/. 

17.  ll^*  +  10:?;'y  +  393^^V2+205;/  +  44y*. 

Factoring  by  Parts. 

§  22.    To  factor  an  expression  which  can  be  reduced  to 
the  form  a  X  F(x)  +  b  Xf(x). 


(L 


FACTORING.  101 


When  the  expression  is  thus  arranged,  any  factor  com- 
mon to  a  and  h,  or  to  F{x)  and /(a;),  will  be  a  factor  of  the 
whole  expression.  The  method  about  to  be  illustrated  will 
be  found  useful  in  cases  where  only  one  power  of  some  let- 
ter is  found. 

Examples. 

1.  Y^^qXqy  acx^  —  ahx —  hc^x-^lp-c. 

Here  we  see  that  only  one  power  of  a  occurs,  and  we 
therefore  group  together  the  terms  involving  this 
letter,  and  those  not  involving  it,  getting 
a  {cx^  —  hx^  —  bc^x  -{-Ifc 

=  ax  (ex  —  b)~be  (ex  —  b)~  (ax  —  be)  (ex  —  b). 

2 .  Factor  m^  x"^  —  mna'^  x  —  mnx  -f-  ri^  a^. 

Here  we  observe  that  a  occurs  in  only  one  power  (a^). 
Therefore,  w^e  have 
—  a^  (mnx  —  n^)  +  I'n?  x^  —  mnx 
=  —  no!-  (nix  —  n)-\-  mx  (mx  —  71) 
=.  (mx  —  n)  (mx  —  na^). 

3.  Factor  2:^'  +  4aa;  +  3Z'^  +  6a^. 

Here  we  observe  that  the  expression  contains  only  one 
power  of  both  a  and  b.  We  may,  therefore,  collect 
the  coefficients  in  either  of  the  following  ways : 

a(4.x  +  Qb)  +  (2x'  +  ?>bx), 
or  5 (3 a;  4-  6a)  +  (2x^  +  4.ax). 
Now  the  expressions  in  the  brackets  ought  to  have  a 
common  factor ;  and  we  see  that  this  is  the  case. 
Hence, 
a(4:X  +  6b)  +  (2x'+Sbx) 

=  2a(2x+3b)  +  x(2x  +  Sb) 

=  (2x  +  3b)(x  +  2a). 


102  FACTORING. 


4.    abxy  +  IP'y^  -\-  acx  —  & 

—  aihxy  +  ex)  +  Z^^y^  —  & 

-=^  ax(hy  +  ^)  +  ihy  +  ^)  (%  —  c) 
^  {hy  +  c){ax  +  hy  —  c). 

=  —  5 {y'  -  2ay  +  a')  +  /  -  2ay'  +  a?y 
-=-h{f  ~2ay  +  a')  +  y{y'^  ~2ay  +  a') 

=  (y~'b){y-a)\ 

6.  2^^y  +  2  hx^  ~  bx^y  +  4 ahx'^y  —  ^^y^  +  4  a^y^ 

—  2  ahxy'^  —  2  ay^ 
=  h (2x^~x^y  +  ^ax^y  ~  2axy'^)  +  2ri;^y  —  ^r^y^ 

+  4a:ry^  — 2a3/^ 
=  Z>a;(2:r^  --  ^r^y  +  ^axy  —  2a?/^) 

+  y  (2rr^  —  :?;^y  +  4  axy  —  2  ay^) 
=  (y  +  hx)  (2x^  —  x'^y-{-4:  axy  —  2  ay"^). 

And  2a;^  —  :?;^y  +  4aa;y  —  2ay'^ 
=  a(4:xy  —  2y^)  +  2c(^  —  ^^y 

—  2ay  {2x  —  y)  +  ^'^  i2x~y) 
=  {2ay  +  x'){2x-^y). 

7.  ^'+(2a~^)a;'-(2a^-a'):r-a'^ 

=  ^  (—  rr'^  —  2  aa;  —  a^)  +  ^r^  +  2a:2;^  +  a?x 
=^  —  h{x-\-ay  +  x(x-\-  ay 
-={x-h){x  +  a)\ 

8.  px^  —  {p  —  q)x'^-\-{p  —  q)x  +  q 

=  q(x^~x  +  l)+px^~px'^+px 
=  q(x'^  —  x+  l)+px(x'^~x  +  l) 
=  lpx+q)(x''-x+l). 

Ex.  36. 

1.  x'^y  —  x'^z  —  y'^  +  yz.  3.    :r^2;^  +  a:r^  — a^z;'^  — a'. 

2.  abxy +  b'^y'^  + acx  — c"^.        4.    2^;^  —  aa;  —  4^a;  + 2a5. 


FACTORING.  lOS 


5.  x'  +  2hx  +  Zax  +  Qab.         8.    ^x'+l2ax+lQihx+lbab. 

6.  x'~h''x'-a'x  +  o?h\  9.    a^  +  {ac~h'')x'+hcx\ 

7.  x''-o?x''-h''x^  +  o^h\       10.    a^  +  {ac~h'')x''-hcc(^. 

11.  ahx^  +  (ac  —  hd)x'^  —  {af+cd)x  +  df, 

12.  j)^'^  —  (p  +  q)x^  -\-  {p-{-  q) X  —  g'. 

13.  d'  +  ab  +  2ac~2b''+lhc  -~^c\ 

14.  r'+(a+l).T'  +  (a  +  l)r?;  +  a. 

^  15 .  Ttipx^  +  (m^'  —  77p)  x^  —  {inr  +  ^$')  x  +  n?\ 

16.  :l^^  —  (a  +  ^  +  c?) o;^  +  (ab  -{-bc-\-  ac)  x  —  abc. 

17.  x^  +  {a  —  b~c)x^~  {ab  —  bc  +  cd) x  +  abc. 

18.  x^  -{-{a-\-b  —  c)  x^  —  (5c?  —  ca  —  ab)  x  —  ahc. 

19.  a!' 01?  —  a^x^y  —  a^xy  +  c^y'^—  ax^yz  +  0:^2;  —  ^^2;  +  ay'^^;. 

20.  a^  bx^  +  a^^^^y  +  acdxy  +  5c(iy^  —  aefxz  —  ^e/ya;. 

21.  a^x^  —  a(b  —  c)x'^  +  c{a  —  b) x  +  c^. 

22 .  ??ia;^  —  n:r^ y  +  r:r^  2;  —  mxy'^  +  ny^  —  ry^ z. 

23.  amri;^  +  (m5y  —  nay  +  mc2;)  :r  —  nby'^  —  ^ic^ya;. 

24.  {am  —  &cm)  r^^  +  {am  —  ben)  x-\-  an-{-  nax. 

25.  db'^ &  —  y^&xy  —  d^&yz  +  c^xy'^z  —  a^b'^zx-]-  b'^x^yz 

-\-a^z^xy  —  x^y'^z^. 

26.  x^—m'^x^—{n—n^)x^  +  {m^n—m^n^)x^  —  a{x'^-\-n^—n). 

27.  1  -  (a  -  1):?;  -  (a  -  ^)  +  1)^;^  +  (a  +  5  -  c?)a;^ 

—  (5  +  c)  a;*  +  cx^. 

28.  a^x^  —  a^{b  —  c+d)x'^y  —  {abc  —  abd-\-acd)xy^+bcdy^. 

29.  m^npx^  —  {n^p  —  m^?^'^  —  m^pq)o(? 

—  {n?  +  71^5'  —  77^^  92^)  :z7  —  n^q. 

30 .  m'^/)^  :?;^  +  ni^p'^  x^  —  {p'^n^  —  (f  m?)  x^ y^ 

—  {p^  n^  —  (f  m^)  x^  y^  —  (7^^  ^^  +  ri'^  (fx)  y^. 


104  FACTORING. 


§23.  Sometimes  an  expression  whicli  does  not  come 
directly  under  the  preceding  form  may  be  resolved  by  first 
finding  the  factors  of  its  parts. 

Examples. 

1 .  ahx^  +  ^i^^y^  —  ^^  ^y  —  ^^  ^y • 

Here,  taking  ax  out  of  the  first  and  third  terms,  and 
by  out  of  the  second  and  fourth  terms,  we  have 

ax  (bx  —  ay)  —  by  (bx  —  ay)^ 
and  hence  {ax  —  by)  (bx  —  ay). 

2.  x^  —  (a  +  b)x^  +  {d'b  +  ab'')x~a^h\ 
Here,  taking  the  first  and  last  terms  together,  and  the 

two  middle  terms  together,  we  have 
(x^  +  ab)  {x^  —  ab)  —  {a  +  b)x^  +  ab  {a  +  b)x 
—  (x^  +  ab)  (x^  —  ab)  —  {a-\-b)  x(x^  —  ab) 
=  (x^  —  ab)  [x'^  +  ab~(a-\-b)  x\ 
=  (x^  —  ab)  (x  —  a)(x  —  b), 

3.  X^'^  —  4:X'^  +  3 

^  ^m|<^2m  _  1)  _  3(^»*  _  1) 

=  x'^lx"'  +  l)(x'^  —  1)  —  S  (x""'  —  1) 
==  (^rf  —  1)  [^™  (^"*  +  1)  —  3]. 

Ex.  36. 

1.  a^  —  ab  ~\- ax  —  bx.  7.  a^ — b'^-j-ax—ac — bx~{-bc. 

2.  abx^-\-Wxy—(j^xy—ab\f.  8.  a^  +  (1  +  a) ab  +  V^, 

3.  x^  +  «^^  —  dx  —  a*.  9.  x^  -\-^xy  {x^  —  y'^)  —  y*. 

4.  a^x-{-2a'^x'^-\-2ax^-{-x^.  10.  x^ —  y^-\- x"^ -{-xy-^y'^. 

5.  acx'  +  (ad-bc)x-bd.  11.  2b  +  (b' -  4:)x-2bx\ 

6.  25x'-6x'  +  x''-l.  12.  x' +  3x^-4:. 


FACTORING.  105 


13.  p'-p'q  —  2pq^  +  2q\       20.    a^  —  4:ah'' +  ?>h\ 

14.  a^  +  a'  —  2.  21.    a''"  —  3  a"^ c'^  +  2 c?-^ 

15.  ?>a^b'-2ah''-~l,  22.    ax^  -  (a^  +  h) x"  +  h\ 

16.  if  —  2>y  +  2.  23.   35a;'^— Ga'^r*^— 9a\ 

17.  2a'-a'Z)-a^'  +  2Z)l      24.    a' ^)2+ 2 a^c^— a' c^- 5^ cl 

18.  5'"*  +  Z)''"  —  2.  25.    arn?--ah''  +  h''m  —  m\ 

19.  y^"— 23/'"2"— 2?/V"+c'^    26.    i— 6a2  +  27a^ 

27.  (:i'-3/7  +  (l-n^  +  y)Cr-y)2;-2l 

28.  24m'  — 28m'n  +  6mn'  — TtiI 

29.  rr'"+"  +  :^;"3/'*  +  a:"*3/"'  +  3/"'+^ 

30.  rr'*  +  207^2/  —  o^o(^-{-  x^'if'  —  2aa;3/^  —  y*. 

Application' OF  the  Theory  of  Divisors. 
§  24.    By  Theorem  I.  we  prove  that 

x^  —  a"^  is  divisible  by  a;  —  a  always^ 

x^  —  a"  is  divisible  by  :^  +  a  when  n  is  even^ 

x"^  +  a**  is  divisible  by  rr  +  a  when  9^  is  odd. 

By  actual  division  we  find  in  the  above  cases : 

^— !^  r=  x''-'  +  x^-'a  + +  m"-'  +  a«-^  (1) 

5!+_^  =  :r"-^  -  x^'-'a  + -  xa""-'  +  a""^  (3) 

Examples. 

1.    Resolve  into  factors  x^  —  y'. 

Here  x  —  y  is  one  factor,  and  by  (1)  the  other  is 
x'  +  xy  +  y''. 


106  FACTOEING. 


2.  Resolve  a^  +  (b  —  cf. 

Here  a  +  (^  —  (?)  is  one  factor,  and  by  (3)  tlie  other  is 
a'-a(b-c)  +  (h-c)\ 

3.  Resolve  rr^^+1024y^^ 

This  equals   (x^f -\-[(2yyY,   one    factor   of  which   is 
x^  -{-  (2y)^  and  by  (3)  the  other  factor  is 
(xj  -  ixy{4:2/)  +  (xji^yj  -  x\AyJ  +  (Aff 
=  x''  -  4:x'y'  +  IQx'y'  -  64.x' f  +  256/. 

4.  Resolve  (x  —  2?/)^  +  (2^  —  3/)^  into  factors. 
Here,  by  (3),  we  have 

=  {x-27/y-(x-  2y)  (2a;  -  y)  +  (2a;  -  y)l 
.*.  the  factors  are  S  (x  —  y)  (7 x"^  —  13:ry  +  7y^). 

5 .  Resolve  rr^  +  rr^y  +  ^^3/^  +  ^^y^  +  ^y*  +  3/^. 

By  (1)  we  see  that  this  equals 

x^  —  y^_  {x'  +  'i/'){x^  —  'i^) 
x—y  ^~y 

=  (^+y)  (^'  -  ^y+y'')  (^' + ^y+y')- 

6.  Resolve  r^;"  —  rr^^a  +  r^^a^  —  x^a^  +  ^^a*  —  :r®a^  +  ^^a^ 

—  x*'a}-\-  x'  a®  —  o;^  a®  +  :ra^®  —  a}^. 
This  equals 

:i;  +  a  x-{-a 

_  (x^  +  g')  (^'  -  ^')  (^  +  a') 

x-\-a 


FACTORING.  107 


Ex.  37. 

Factor  the  following : 

1.  x^-y^',  T'~\\  rr'  +  8;  8a'-27:r^  8  +  a'a;^ 

2.  a;^-a^^  27a^-64;  a^^  -  Z;^  rr^«-32y^. 

3.  Find  a  factor  which,  multiplied  by 

a'  +  a'^  +  a'Z^'  +  ab^  +  ^^^  will  give  a^  -  Z^^ 

4.  By  what  factor  must  x^—  4:x'^y-{-16xy'^~(j4:y^  be  mul- 

tiplied to  give  x^  —  256  2/^  ? 

5.  Factor  x'^  +  x^y  +  x^y"^  +  ^*y^  +  ^^y*  +  ^^3/^  +  ^3/^  +  3/^- 
Find  the  factors  of  the  following  : 

6.  (By'~2xy~(Sx'-2yJ;  a'~l6b\ 

7.  x^  —  y^  —  x(x'^  ~  y'^)  +  3/ (a;  —  y)^ 

8.  Z>  (x^  —  a^)  +  <^^ (^^  —  «0  +  <^^  (^  —  «). 

9.  b  {m^  +  a^)  +  am  {m^  —  a^)  +  a^  (m  +  a). 

10.  x^-~y^  +  2xy(x'  +  x'y''  +  y'). 

11.  (a'  — Z)c)'  +  8Z>^c^  rr*"*  — a'». 

12.  rr^-3a2;2  +  3a=':r-a'  +  Z>l 

13.  (ri;'  +  Sf)(x  +  y)  —  Qxyix"  —  2^y  +  4/). 

14.  8:r'  — 6a:y(2rr  +  3y)  +  27yl 

15.  l  —  2x+4:x^-^x\ 

16.  a^  +  a^Z^c  +  a^b'^c'  +  a^b^&  +  ab^'c'  +  ^^^c^ 

§25.  The  principles  illustrated  in  Chap.  II.  may  be  ap- 
plied to  factor  various  algebraic  expressions,  as  in  the 
following  cases : 


108  FACTORING. 


Examples. 

1.  Find  tlie  factors  of 

{a  +  h  +  c)  {ah  +  hc  +  ca)  —  (a  +  h)  (h  +  c)(c  +  a). 

1.  Observe  that  the  expression  is  symmetrical  with  respect 
to  a,  5,  c. 

2.  If  there  be  any  monom>ial  factor,  a  must  be  one.  Put- 
ting a  =  0,  the  expression  vanishes ;  hence,  a  is  a 
factor,  and,  by  symmetry,  h  and  c  are  also  factors. 

Therefore,  ahc  is  a  factor. 

3.  There  can  be  no  other  literal  factor,  because  the  given 
expression  is  of  only  three  dimensions,  and  ahc  is  of 
three  dimensions. 

4.  But  there  may  be  a  num^erical  factor,  m  suppose,  so 
that  we  have 

{a-Yh-\-c)  {ah-\-hc-\-ca)  —  {a-\-h)  (h+c)  (<?+a)  =  m.ahc. 
To  find  m,  put  a  =  ^  —  c  =  l  in  this  equation,  and  m  =  1. 
Therefore,  the  expression  =  ahc. 

2.  Resolve  a\h  —  c)  +  h\c  —  a)  +  c\a  —  h). 

1.  For  a  =  0  this  does  not  vanish  ;  hence,  a  is  not  a  fac- 
tor, and,  by  symmetry,  neither  is  h  nor  c. 

2.  Try  a  hinom^ial  factor  ;  this  will  likely  be  of  the  form 
h  —  c]  put  h  —  c  =  0]  that  is,  h~c  in  the  given  ex- 
pression, and  there  results 

a^  {c  —  c)  +  c^  (c  —  a)  +  c^{a  —  c),  which  =  0. 

Therefore,  h  —  c  is  a,  factor,  and,  by  symmetry,  c  —  a  and 
a  —  h  are  factors.  Since  the  given  expression  is  only 
of  three  dimensions,  there  can  be  no  other  literal  fac- 
tor ;  but  there  may  be  a  numerical  factor,  m  suppose, 
so  that 
a^  (h-c)  +  h'  (c-a)  +  c\a~h)  =  m  (a-h)  (h-c)  {c~a). 


FACTORING.  109 


To  find  the  value  of  m,  give  a,  b,  c  in  this  equation,  any 
values  which  will  not  reduce  either  side  to  zero  ;   let 
a^lj    h  =  2,    c=^0,    and  we   have   2=:m(— 2),  or 
m  =  —  1 ;  so  that  the  given  expression 
=  —  {a  —  h)(b  —  c)(c  —  a),  or  (a  —  h){b  —  c)  (a  —  c). 

3.  Kesolve 

a'  {h  +  c^)  +  h\c+  o?)  +  c\a  +  b')  +  abc  (abc  +  1). 

Here  we  see  at  once  that  there  is  no  Tnonomial  factor. 
Put  b-\~  c^  =  0,  that  is,  b  =  —  c^,  and   the  expression 

becomes 
a\-~c'  +  c')-c'(c+a')  +  c'{a+c')-c'a(-c'a+ll 

which  =  0. 
.'.b-j-c^  is   a   factor;    and,  by  symmetry,   c-{-a^  and 

a-^-b"^  are  also  factors ;    and  proceeding  as  in  former 

examples,  we  find  m  =  1. 
.*.  the  expression  ={b  -{-  c^)  {c  +  a?)  {a  +  b'^), 

4.  Resolve  into  factors  the  expression 

(a-by  +  {b  -  cf  +  {c-  a)\ 

As  before,  we  find  that  there  are  no  monomial  factors. 
Let  a  —  5  =  0,  ora  =  &;  and,  substituting  b  for  a,  the 

expression  becomes  zero.     Hence,  a  —  5  is  a  factor  ; 

by  symmetry,  b  —  c  and  c~  a  are  factors.     Hence, 

the  factors  are  m  (a  —  b)(b  —  c)  {c  —  a). 
To  find  m,  let  a  =  0,  5  =  1,  (?  =  2,  and  we  have 

6  =  2m,  or  7?i  =  3. 
The  factors  are,  therefore,  3  (a  —  5)  (5  —  c)  {c  —  a), 

5.  Resolve  into  factors 

a'  (b-c)-\-  b'(c  --a)  +  c'(a-  b). 
As  before,  we  find  that  there  are  no  monomial  factors. 


110  FACTORING. 


Let  a  —  Z>  =  0,  or  a  =  b  ;  substituting  b  for  a,  the  ex- 
pression becomes  zero.  Therefore,  a  —  b  is  a  factor; 
by  symmetry,  b  —  c  and  c  —  a  are  factors. 

Now  the  product  of  these  three  factors  is  of  three  dimen- 
sions, while  the  expression  itself  is  oifour  dimensions. 
There  must,  therefore,  be  another  factor  of  one  dimen- 
sion. It  cannot  be  a  monomial  factor,  for  the  expres- 
sion has  no  such  factors.  It  cannot  be  a  binomial 
factor,  such  as  a-\-b,  for  then,  by  symmetry,  b-\-c  and 
c+a  would  also  be  factors,  which  would  give  an  ex- 
pression of  six  dimensions.  It  cannot  be  a  trinomial 
factor,  unless  a,  b,  and  c  are  similarly  involved.  For 
instance,  if  a  —  b-\-c  were  a  factor,  then,  by  sym- 
metry, b  —  c-\-a  and  c  —  a-\-b  would  also  be  factors, 
and  the  dimensions  would  be  six  instead  oifour.  The 
other  factor  must  therefore  be  a-{-b-\-c.     Hence,   ». 

a\b  -  c)  +  b\c  -  a)  +  c\a  -  b) 

=  7n(a  —  b)  (b  —  c)  (c  —  a)  (a  -{-  b  -{-  c). 

To  find  m,  put  a  —  0,  5  =  1,  and  c=2,  and  we  have 
—  6  =  6  m ;  therefore,  m  —  —  1. 

Hence,  the  factors  are  —(a—b)(b  —  c)(c—a){a-{-b-{c), 
or  (a  —  b){a  —  c)(b  —  c)(a-{-b-^  c). 

Prove  that 

a'  +  b'  +  c'  +  S(a  +  b)(b  +  c)(c  +  a) 

is  exactly  divisible  by  a  +  5  +  ^,  and  find  all  the 

factors. 
Let  a  -f  5  +  c  =  0,  or  a  —  —  (b-\-c)',    substituting  this 

value  for  a,  we  have 
-{b  +  cf  +  b'  +  c'  +  3bc(b  +  c), 

OT-(b  +  cy  +  ib  +  c)\ 

which  =  0 ;  and  therefore  a  +  b  -\- c  is  sl  factor. 


FACTORING.  Ill 


As  before,  we  find  that  there  are  no  monomial  factors. 

Since  a-^b-^c,  the  factor  already  obtained,  is  of  07ie 

dimension,  the  other  factor  must  be  of  two  dimensions, 

and  as  it  must  be  symmetrical  with  respect  to  x,  y, 

and  z,  it  must  be  of  the  form 
m (a^  +  b^  +  c^)  +  n(ah  +  bc-\-  ca), 

in  which  rfi  and  n  are  independent  of  each  other,  and 

of  a,  Z>,  and  c. 
To  determine  their  values,  put  c  =  0,  so  that 
o?  +  h^  +  c^  +  ?>{a  +  h){h  +  c)  {c  +  a) 

=  {a  +  h  +  c) [m{o?  +  ^'  +  C")  -\-n{ah  +  hc  +  ca)] 

becomes 
a?  +  h^  +  ^ah{a  +  h)  =  {a  +  h)[m{a''  +  h'')  +  nah\ 

But 

a^-\-P  +  Sah{a  +  h)  ==  (a  +  h)\ 

.-.  (a  +  hf  =  {a  +  b) [m {p?  +  }f)  +  nah\ 

:.(a-\-  by  =  m (a^  +  b"^)  +  nab. 

That  is, 

^2  _j_  j2  _|_  2ab  =  771(0"  +  b')  +  nab. 

Now  this  is  true  for  all  values  of  a  and  b. 

.*.  m—  1  and  n  =  2. 

.-.  a'  +  ¥  +  c'  +  3(a  +  b)(b  +  c)(c  +  a) 

=  (a  +  b  +  c)[a''  +  b'  +  c'  +  ^lab  +  Z^(?  +  ca)] 

=  (a  +  b  +  c){a  +  b  +  cy 

==(a  +  b  +  cf. 

7.    Simplify 

a(b  +  cf+b(a  +  cy  +  c(a  +  by--(a  +  b)(a-c)(b~c) 

-  (a-b)(a-c)(b  +  c)  +  (a-b)(b-  c)(a  +  c). 
Let  a  =  0,  and  the  expression  becomes 
be'  +  cb'  +  ^^(5  -  c)  -  Z>c(^  +  ^)  -  ^^(^  -  c), 

which  equals  zero ;  therefore  a  is  a  factor ;  by  sym- 
metry, b  and  c  are  also  factors. 


112  FACTOEING. 


The  expression  is  of  three  dimensions,  and  abc  is  of 
three  dimensions,  there  cannot  therefore  be  any  other 
literal  factor. 

Hence  the  expression  equals  mahc. 

To  find  m,  let  a  =  Z>  =  c  =  1,  and  we  have 
4  +  4  +  4  =  m;  m  =  12. 

.'.  the  expression  =  12ahc, 

In  the  preceding  examples  the  factors  have  been  linear, 
but  the  principle  applies  equally  well  to  those  of  higher 
dimensions.     (See  Th.  II.  Cor.) 

8.  Examine  whether  x'^-\-l  is  a  factor  of 

Let  x^  +1^=0,  or  ^'^  =  —  1,  and  substituting,  the  ex- 
pression vanishes ;  therefore,  :?;**  +  1  is  a  factor. 

9.  Examine  whether  a^  +  ^^  is  a  factor  of 

2a''  +  a'h  +  2a^h''  +  ah\ 
Let  o?  -{-h^  =  0,  or  a^  =  —  H^.     Substituting,  we  have 
2b'-ah'-2h'  +  ah\ 
which  =  0,  and  therefore  o?  +  Z>Ms  a  factor. 

10.    Prove  that  o?  +  ^^  is  a  factor  of 

w'+a'h  + 0^1^  +  0.^1)^  + ah' +  h\ 
Let  a^  +  h^  —  0,  or  a^  =  —  h^.     Substituting,  we  have 
-a'P-  al'  -  5^  +  a^Z>^  +  ah'  +  l\ 
which  =  0,  and  therefore  d^  +  Z>Ms  a  factor. 


Ex.  38. 

Kesolve  into  factors : 

2.    hc(b  — c)  — ca{a  — c)  ~  ah(h  —  a). 


FACTORING.  113 


5.  {a  +  hy--{h  +  cf  +  {c-a)\ 

6.  a(h  —  cy  +  h{c-ay  +  c{a~h)\ 

7.  {a  +  h  -\-  c)  (ah  -{-he  -{-  ca)  —  ahc. 

8.  a'  (c  -  h')  +  ^-^  (a  -  c^)  +  ^'  (^  -  a^)  +  ahc  (ahc  -  1). 

9.  a\h  +  c)-\-h\c  +  a)  +  c'(a+h)  +  2ahc. 

10.  (a  —  ^)  (c  -  h)  (c  -  XO  +  (^  -  c)  (a  —  h)  (a  —  ^) 

+  (^-a)(Z)-A)(^-^). 

11.  x^y""  +  ^r'y*  +  :r'2'  +  x'z'  +  7/'^'  +  ?/'2*  +  2x'y'z\ 

12.  (a-Z>)^  +  (^)-c7  +  (c-a)l 

13.  ah(a  +  h)  +  hc(h  +  c)  +  ca(c  +  a)  +  (a'  +  ^'  +  c')- 

14.  a'  (c  -  P)  +  h\a-  c')  +  c'(h~  a')  +  ahc  (a'  h'  c'-l). 

15 .  x'(y'~  z')  +  7/  (z'  -  x')  +  ^'^  (:?;^  -  y'). 

16.  !^*  +  y*  +  ;s*  ^  2:r'?/'  "  27/ z'  -  2z'x'. 

17.  (^  —  c)  (x  —  5)  (rt'  —  c)-{-(G  —  a)(x  —  c)  (x  —  a) 

-{-  {a  —  h){x  —  a){x  —  h). 

18.  {a  +  hy  +  (h  +  cy  +  {G  +  af 

+  '^{a  +  2h  +  c){h  +  2c  +  a){c  +  2a  +  h), 

19.  Show  tliat  a^  -\-d!'W  —  ah'^  —  ¥  has  a^  —  5  for  a  factor. 

20.  Show  that  (x  +  ?/)^  —  x'^  —  / 

=  1xy{x  +  7j)  (x"  +  xy  +  y^. 

21.  Examine  whether  x^  —  5  a;  +  6  is  a  factor  of 

22.  Show  that  a  —  5  +  c  is  a  factor  of 

a\h  +  c)-  h^c-i-a)  +  c\a  +  h)  +  ahc. 


114  FACTORING. 


23.  Show  that  (2^  +  3^  is  a  factor  of 

and  find  the  other  factor. 

24.  Find  the  factors  of  a^  {h  —  c)  +  h\c— a)  -\-  c^  {a—h). 


Factoring  a  Polynome  by  Trial  Divisors. 

§  26.    To  find,  if  possible,  a  rational  linear  factor  of  the 
polynome 

x^  +  hx""-^  +  ex''-''  + -{-hx  +  h 

in  which  h,  c,  ,  A,  h,  are  all  integral,  substitute  succes- 
sively for  X  every  measure  (both  positive  and  negative)  of 
the  term  h,  till  one  is  found,  say  r,  that  makes  the  poly- 
nome vanish,  then  x  —  r  will  be  a  factor  of  the  polynome. 

Examples. 
1.    Factor;r^  +  92;'+16:ir  +  4. 

The  measures  of  4  are  ±1,  ±2,  and  ±  4.  Since  every 
coefficient  of  the  given  polynome  is  positive,  the  pos- 
itive measures  of  4  need  not  be  tried.  Using  the 
others,  it  will  be  found  that  —2  makes  the  poly- 
nome vanish.     Thus, 

1         9         16        4 
-2    -14     -4 


-2 


17  2;       0 

Hence,  the  factors  are  {x  -\-  2)  (x^  -\-l  x  -{-  2). 

The  labor  of  substitution  may  often  be  lessened  by  ar- 
ranging the  polynome  in  ascending  powers  of  x,  and  using 
the  reciprocals  of  the  measures  of  Ic  instead  of  the  measures 
themselves.  Should  a  fraction  occur  during  the  course  of 
the  work,  further  trial  of  that  measure  of  h  will  be  needless. 


FACTORING. 


115 


Factor  x^ -I0x^-^?>x  +  60. 

The  measures  of  60  are  ±  1,  rb  2,  i  3,  ±  4,  zh  5,  etc. 

Neither  +  1  nor  —  1  will  make  the  polynome  vanish. 

Try  2 ;  thus, 


1 

60 

-63 

30 

-10 

1 

2 

30 

-16} 

A  fraction  occurring,  we  need  go  no  further.  —  2  will 
also  give  a  fraction,  as  may  easily  be  seen.  Next 
try  3 ;  thus, 

60         -63         -10         1 
20 


20 


14i 


A  fraction  again  occurring,  we  may  stop, 
also  give  a  fraction.     Next  try  4 ;  thus, 


-  3  will 


60 

-63 

-10 

1 

1 

15 

-12 

4 

15 

-12 

-5i 

-  4  will  also  give  a  fraction.     Next,  trying  5,  we  find 
it  fails,  and  we  then  try  —  5  ;  thus, 


1 

60 

-63 

-12 

-10 
15 

1 
-1 

5 

12 

-15 

1; 

0 

The  remainder  vanishes.     The  factors  are,  therefore, 

{x  +  b){x'-lbx+l2). 

§  27.  When  k  has  a  large  number  of  factors,  the  number 
that  need  actually  be  tried  can  often  be  considerably  les- 
sened by  the  following  means  : 


116  FACTORING. 


For  X  substitute  successively  three  or  more  consecutive 

terms  of  the  progression  ,  3,  2,  1,  0,  —1,  —2,  —3,  

Let ,  h^,  ^2,  ^1,  ^,  ^-1,  ^-2,  ^-3, ,  denote  the  correspond- 
ing values  of  the  polynome  ;  and  let  r  denote  a  measure  of 
Ic  positive  or  negative. 

The  substitution  of  r  for  x  need  not  be  tried  unless  r  —  1 

measure  Ic^,  r  — 2  measure  ^2,  ,  and  also  r  +  1  measure 

1c_i,  7- +  2  measure  lc_2,  If  no  measure  of  h  fulfil  these 

conditions,  the  polynome  will  have  no  linear  factor. 

If  p  denote  a  positive  or  arithmetical  measure  of  k,  the  preceding 
criterion  may  be  conveniently  expressed  as  follows : 

1.  The  substitution  of  +p  for  x  need  not  be  tried  unless  jo  —  1 

measure  \,  p  —  2  measure  Jc^, ,  and  also  p  +  1  measure  k.-^,  p+  2 

measure  p_2,  

2.  The  substitution  of  — ^)  for  x  need  not  be  tried  unless  p  +  1 

measure  ^j,  p  +  2  measure  k^,  ,  and  also  p  —  1  measure  k_i,  p  —  2 

measure  k_2,  

In  trying  for  measures,  the  signs  of  ^2,  Jci,  k,  ,  may 

evidently  be  neglected. 

If  kt  vanish,  t  positive  or  negative,  then  x—  t  will  be  a 
factor  of  the  polynome,  and  should  be  divided  out  before 
proceeding  to  test  for  other  factors. 

Examples. 

1 .    Eind  the  factors  of  ^'  —  10  ^'  —  63  ^  +  60. 

Here  ^=-60,  ^i=— 12,  ^2-=-98,  /;_i  =  112,  ^_.2-138. 
Tabulating  the  trial  measures,  we  get 


k; 


98 

1, 

9 

12 

2 

3, 

4, 

60 

3, 

4, 

5, 

6, 

112 

4, 

7. 

138 

10,     12, 


FACTORING. 

98 

7, 

12 

4, 

6, 

60 

3, 

4 

5,  6,  1 

112 

4, 

138 

1, 

3, 

117 


10, 


In  the  upper  or  positive  table,  no  measure  of  60  gives 
a  full  column  ;  hence,  no  positive  integer  substituted 
for  X  will  make  the  given  polynome  vanish. 

In  the  lower  or  negative  table,  5  is  the  only  measure 
of  60  that  gives  a  full  column ;  hence,  —  5  is  the 
only  negative  integer  that  need  be  tried  for  x.  Sub- 
stituting —  5  for  X,  the  polynome  vanishes ;  hence, 
^  +  5  is  a  factor  of  x^  —  lOx'^—  63rr  +  60. 

In  constructing  the  above  tables  it  is  evident  that  12 
is  the  highest  measure  of  60  that  need  be  tried  in 
the  upper  table,  for  the  next  measure,  15,  would 
give  14  as  a  trial-measure  of  12  (the  absolute  value 
of  ^_i),  and  higher  measures  would  give  higher  trial- 
measures.  Similarly,  10  is  the  highest  measure  that 
need  be  tried  in  the  lower  table. 

Since  it  can  make  no  difference  in  the  full  columns 
which  of  the  lines  of  measures  is  made  the  basis  from 
which  to  construct  these  columns,  it  will  be  found 
best  to  construct  the  tables  by  the  measures  of  that 
one  of  the  ^'s  which  has  the  fewest  number  of  them. 

Find  the  factors  of  x'  +  12:?;^  -  iOx'  +  67  x  -  120. 
^--120,  ^i--80,  h  =  -^^,  yl'_i=--238. 
Selecting  the  measures  of  34  for  trial-measures,  and 
tabulating,  we  get 


34 

1 

80 

2 

120 

238 

3 

2,     17,     34, 


17,    84, 
16, 
15, 
14, 


118 


FACTORING. 


Here,  in  the  only  column  that  is  full,  15  stands  in  the 
line  of  120,  the  absolute  value  of  h,  and  as  the  col- 
umn is  decreasing  the  sign  of  the  15  must  be  minus; 
hence,  the  only  measure  of  k  that  need  be  tried  is 
—  15.     On  substituting  —  15  for  x,  we  get 


-120 

67 

-40 

12 

1 

-1 

8 

-5 

3 

-  1 

15 

-8 

5 

-3 

1; 

0 

Hence,  the  only  linear  factor  of  the  given  polynome  is 
X  +  15,  and,  as  is  seen  from  the  substitution,  the 
other  factor  is  x^  —  '^x;^  -\-bx  —  8. 


Factor  x"^  —  21  x^  +I4cx+  120. 
^-120,  ^1-108,  ^2  =  56,  h_,  =  SO. 


56 

1,  2,  4,  7,  8,  14,  28,  56, 

4,  7,  8,  14,  28,  56, 

108 

2,  3,    9, 

3,  6,     27, 

k;   120 

3,  4,    10, 

2,5, 

80 

4,5, 

1,4, 

The  positive  or  increasing  columns  give  3  and  4  to  try  ; 
the  negative  or  decreasing  columns  give  —2  and  —5. 
Using  these  in  order,  we  get 


x—'^  is  a  factor. 


120 

14  ■ 

-27 

0 

1 

1 

40 

18  - 

-8  - 

-1 

8 

40 

18 

-8  - 

-1; 

0 

1 

10 

7 

1 

4 

10 

7 

1; 

0 

-1 

-5 

-1 

2 

5 

1; 

0 

-4  is  a  factor. 


rr+2  is  a  factor, 


and  there  remains  :r+5,  a  factor. 
Hence,  the  factors  are  {x  —  Z){x  —  4)  (x  +  2)  (^  +  5). 


FACTOHING. 


119 


4.    Factor  x*"  —px^  -\-  {q  —1)  x"^  -\-px  —  q. 

Jc=—q,Ici    =l~p  +  (q-l)+p  —  q  =  0, 
h_,  =  l+p  +  {q-l)--p-q^O. 
Since  both  k^  and  k_i  vanish,  the  polynome  is  divisible 
by  both  X  —\  and  x-\-l. 


1 

1 

-p 
1 

-^3  +  1 

p 
i-'P 

-q 

!Z 

1 

1 

-p+l 
-1 

a-p 
+p 

-1 

0 

1 

-p 

?; 

0 

Hence,  the  other  factor  is  x^  —px-{-  q. 
5 .    Factor  x'  +  2 ax^  +  {a"  +  a)x''  +  2a^x  +  a\ 

h  ^  a\  h    =  1+  2  a + {o?+  a) +  2  a^+  a'  =  (a+l)\ 

The  positive  measures  of  Ic  are  1,  a,  a^,  a^.  Of  these  1 
may  be  rejected  at  once,  since  neither  Jci  nor  ^2  van- 
ish ;  and  a^  and  a^  may  also  be  rejected,  since  ki  or 
(a+  1)^  is  not  divisible  by  either  a^  ±  1  or  a^  rb  1. 
But  Jci  is  divisible  by  a-\-l,  and  Jc^i  is  divisible  by 
a—l]  thus,  we  need  try  the  substitution  of  only 
—  a  for  X. 


1 

2a 

a' 

+  a 

2  a' 

a? 

—  a 

-a' 

-a' 

-a' 

1 

a 
—  a 

a 
0 

-a' 

0 

1 

0 

a; 

0 

acto] 

'S  are 

^x- 

+  a) 

^(xH 

-  a 

). 

6,    Factor  ^^  —  (a  +  6')  x'^  +  (^  +  ac)  x  —  be. 
Ic     =-~  he, 

Jci    ==      l—(a-\-e)-\-(b-\-ae)—he  =  l—a-j-b—e-^-ac—he, 
Jc^i  =  —  l~(a-]-c)—(h^-ae)—be  =  —  (l+a+5+c+«<?+^^)- 


120 


FACTORING. 


The  factors  of  ^i,  other  than  1,  are  h  and  c.  ki  is  not 
divisible  by  either  h±:l  nor  by  C+  1.  However,  Ic^ 
is  divisible  by  <?  —  1,  and  h_i  is  a.t  the  same  time 
divisible  by  <?  +  1 ;  hence,  we  need  try  the  substitu- 
tion of  only  c  for  x. 

1         ~{a  +  c)         {h  +  ac)         ~hc 
c  —  ac  he 


\         —a  h  ] 

Hence,  the  factors  are  {x  —  c)  {x^  —  ax-\-h). 


Ex. 

39. 

1. 

a^-9a^  +  16a-4. 

15. 

2. 

^r'- 9^' +  26^ -24. 

16. 

3. 

^^-7;r^  +  16:^-12. 

17. 

4. 

^^-12:^;  + 16. 

18. 

5. 

x^-\-^x^  +  bx  +  ?>. 

19. 

6. 

x'+^x'+l0x'+l2x+^. 

20. 

7. 

x^-^x  +  2. 

21. 

8. 

x^  +  2x^  +  ^. 

22. 

9. 

m^—  3  w?n + 4  mv?—  2  n^. 

23. 

10. 

x^ +  2x^  +  2. 

24. 

11. 

m^—  5  w?n + 8  mn^~  4  nl 

25. 

12. 

h^+h''c  +  1hc'+Z%c\ 

26. 

13. 

m^~4:mn^  +  Sn^. 

27. 

14. 

a'-7a'b+2SaP-16b\ 

28. 

29.    x'-lSx'+USx'- 

•30.    ^*-9:r'y  +  20 

x''^/ 

:r^- 11^^ +  39:?; -45. 
x^  +  bx''+7x  +  2. 
a^-3a^- 193a +195. 
p'  -  Sp'  -  6j9  +  8. 

a^-Qa'"" +110"^ -6. 
a'~4:la'P  +  16b\ 
a'-a'b'-2ab^  +  2b\ 
f  -  ^f  +  6^-4. 

y*-5y^  +  8y^-8. 
a*-2a'+3-a'-2a  +  l. 
a^-\-a^W'~\-ab'--?>b\ 
2a^«_a'«_a"  +  2. 
288a; +252. 
-392:y«  +  18?/^ 


FACTORING.  121 

§28.    To  find,  if  possible,  a  rational  linear  factor  of  the 
poly nome     ^^n  ^  ^^n-i  _^  ^^n-2  _^ +  hx  +  Jc, 

in  which  a,  h,  c h,  h  are  all  integral. 

First  Method.     Multiply  the  polynome  by  a"~\ 

{axy+h(axy-^+ac{axy~^'+ +aJ'-Vi{ax)+a''-^k] 

or,  writing  y  for  ax, 

yu  _|_  lyn-Y  _|_  ^^yn-1  _j_ _|_  ^n-IJ^y  _j_  ^n-1^ 

Factor  this  polynome  by  the  method  of  the  last  article, 
replace  y  by  ax,  and  divide  the  result  by  a''"^ 


Example. 

Factor  3  ^*  +  5  ^'  -  33  x^  +  43  :r  -  20. 
Multiply  by  3^,  and  express  in  terms  of  3  x. 

(3  x)'  +  5(3  xj  -  99  (3  xj  +  387  (3  x)  -  540 ;  or, 

y^  +  53/^  -  99y^  +  387y  -  540. 

Here  ^^=-540,  h   =-1  +  5-99  +  387-540  =  -246, 
Z;_i  ==  1  -  5  -  99  -  387-  540  =  - 1030. 

6,     41,     82,     123,     246. 


246 

540 

1030 


1, 
9 


2 
3, 


3, 
4, 
3, 

41, 


246      3,     6,     41,     (Trying  by  factors  of  246, 

540      2,     5,  instead  of  by  factors  of 

1030      1,  540,  for  convenience.) 

The  only  factors  of  540  in  full  columns  are  4  in  the 
upper  table  and  2  in  the  lower  one ;  hence,  we  need 
try  only  the  substitutions  4  and  —  2. 


1 

-540 

387 
-135 

-99 
63 

5 
-9 

1 
-1 

4 

-  135 

63 

-9 

-1; 

0 

122  FACTORING. 


Hence,  3/  —4  is  a  factor.     The  substitution  of  —2  need 
not  now  be  tried,  since  we  see  tliat  135  is  not  a  mul- 
tiple of  2.     The  other  factor  is,  therefore, 
y'  +  9y'--63y  +  135. 

Replacing  y  by  3r^,  and  dividing  by  27, 
2V(3^  -  4)  (27r^  +  81:^^  -  189^  +  135) 
-=  (3^  -  ^){x^  +  Zx^  -  7^  +  5), 
which  are  the  factors. 

§29.  Second  Method.  Write  7)i  for  "a  measure  of  a," 
and  r  for  ''a  measure  of  Ic,  positive  or  negative " : 

For  X  substitute  every  value  of  t-'^th  till  one,  say  r'-^m', 
be  found  which  makes  the  polynome  vanish ;  then  on'x  —  r' 
will  be  a  factor.  Should  a  fraction  be  met  with  in  the 
course  of  substitution,  further  trial  of  that  value  of  r  -^  m 
will  be  useless. 

Should  Ic  have  more  factors  than  a,  it  will  generally  be 
better  to  arrange  the  polynome  in  ascending  powers  of  x 
and  use  values  of  m  -^  r  instead  of  r  -v-  m,  making  r  positive 
and  on  positive  or  negative. 

To  reduce  the  number  of  trial  measures,  for  x  substitute 
successively  three  or  more  consecutive  terms  of  the  pro- 
gression   ,  3,  2,  1,  0,  —1,  —2,  —3, ,  then  denoting  the 

corresponding  values  of  the  polynome  by  ,  h^,  h^,  h^,  Ic^ 

The  substitution  of  r  for  x  need  not  be  tried  unless  r— ??i 

measure  ^1,  r— 2?7i  measure  Jc^^  ,  and  also  T-\-in  measure 

^_i,  rH-2m  measure  lc_2,  

If  p  denote  a  positive  or  arithmetical  measure  of  h,  this  criterion 
may  be  expressed  as  follows : 

1.    The  substitution  oi  +p  for  x  need  not  be  tried  unless  p  —  77i 

measure  h-^,  p  —  2m  measure  k^,   ,  and  also  p  +  m  measure  k^^, 

p  f  2  m  measure  ^  ..^,  


FACTORING.  123 


2.    The  substitution  of  —  ^  for  x  need  not  be  tried  unless  |)+m 

measure  k^,  p-\-2m  measure  k^,  ,   and  also  p  —  m  measure   ^_i, 

p  —  2m  measure. ^_2, 

It  must  be  remembered  that  here  m  may  be  either  positive  or 
negative,  as  may  also  be  one  or  more  of  the  quantities,  p  -\r'm,  p  —  m^ 
p  -\-  2m,  p  —  2m,  etc. 

Examples. 

1.    Factor36:r'^+171a:^-22r?;  +  480. 

^'-480,        k   =-665,        h   -1408, 
k_,  -  637,         Ic_,  =  920, 

and  7)1  may  have  any  of  tlie  values, 

±1,  ±2,  ±3,  ±4,  ±6,  ±9,  ±12,  ±18,  ±36. 

In  forming  the  table  of  trial-measures,  write  out  the 
measures  of  1408,  that  is,  k^ ;  they  are 

1,  2,  4,  8,   11,   16,  22,  32,  44,  64,  88,  128,  176, 
352,    1408. 

Taking  each  of  these  in  succession,  add  to  it  each  value 
of  7)1  separately.  Should  the  sum  appear  among 
the  measures  of  665,  that  is,  Ic^,  which  are 

1,   5,    7,    19,   35,   95,    133,   665, 

enter  these  measures  of  k.2  and  Jci  in  a  column  in  the 
table,  writing  above  them  the  value  m  used.  How- 
ever, should  the  sum  not  be  a  measure  of  665,  another 
value  of  m  must  be  tried.  When  all  the  values  of 
771  have  been  tried  with  one  measure  of  1408,  another 
measure  must  be  taken  till  all  have  been  used.  This 
having  been  done,  proceed  to  test  which  of  the  col- 
umns can  be  filled  up  with  measures  of  480,  637, 
and  920,  respectively,  these  being  the  values  in  this 
case  of  Ic,  k__i^  k^2- 


124 


FACTORING. 


The  table  will  then  appear  thus : 

4-4,  +6,  +3,  +1,  +3,  +3,  +3, 


^; 


1408 

1, 

1, 

2,  4,  4,  16,  32, 

665 

5, 

7, 

5,  5,  7.  19,  35, 

480 

8,  6,  10, 

637 

7,  13 

920 

8, 

1113 

-2, 

-6,  -3, 

-9,  -3, 

-9, 

-1,  -3, 

_9,  -4,  -6, 

-9, 

-3, 

1408 

1, 

1,  2, 

2,  4, 

4, 

8,  8, 

8,  11,  11, 

16, 

22, 

665 

-1, 

-5,  -1, 

-7,  1, 

-5, 

7,  5, 

-1,  7,  5, 

7, 

19, 

480 

-3, 

-4, - 

-16,  -2, 

6,  2, 

-10,  3,  -1, 

-2, 

16, 

637 

-7, 

-1, 

-1,  -7, 

13, 

920 

-10, 

-4, 

-5, 

10, 

]113 

-7, 

7, 

There  still  remain  five  full  columns,  while  the  given 
polynome,  being  of  the  third  degree,  cannot  have 
more  than  three  linear  factors.  To  reduce  the  num- 
ber of  these  columns,  and,  as  a  consequence,  the 
number  of  trial-measures,  extend  the  table  by  calcu- 
lating ^_3  and  the  corresponding  column-numbers 
for  the  full  columns.  ^_3  =  1113,  and  the  column- 
numbers  are  9,  —13,  —7,  —9,  and  7.  Of  these,  9, 
—  13,  and  —9  must  be  rejected,  not  being  measures  of 
1113.   This  leaves  only  —7  and  7,  to  which  correspond 

Q  Q 

——  and  ■— —  as  the  values  of  —  to  be  tried  in  substi- 
2  16  r 

tution  for  -.     (See  table  above.)     Making  trial  of 

these  two,  the   polynome   is   found   to  vanish  for 

-— -  but  not  for  — —• 
16  2 

The  actual  work  of  substitution  will  be  as  follows. 


FACTORING. 


125 


Arrangement  in  ascending  powers  of  x  : 

480  -22  171  36 

-  720        1113        -  1926 


^3 


2 
-3 


16 


240 
480 


-371 

—  22 
-90 


642; 

171 
21 


-945 

36 
-36 


30 


—  / 


12; 


0 


Hence,  tlie  factors  are  Sx-{-16  and  12:?;^  —  7^  + 30. 
The  latter  factor  cannot  be  resolved,  for  it  does  not 
contain  3x-\-16,  and  the  only  other  factor,  viz., 
3r^  +  2,  left  for  trial  by  the  tables  above,  has  been 
tried  and  has  failed. 

2.    Factor  10 a;^  -  x' (15 ?/  +  4 2)  -  x'  (40?/^  -63/^) 
+  x(60f  +  Uy'z)  -  24:y^z. 

Here  on  =  ±  1,  ±  2,  i  5,  or  i  10. 
Jc  =  -24:y'z. 

y^i  -  10  -  (153/  +  4^)  -  (40y^  -  67/z) 
+  (60y'+16y'z)-24:y'z 
=  10--152j-4:0f  +  60y' 

-2z(2~3y-Sf  +  12y') 
--(5-20)(2-3y-8/  +  123/^). 
^_,-(5  +  2.)(2  +  3y-8y^-122/0, 
as  may  easily  be  found  by  making  the  calculation. 
We  get  at  a  glance  2:2  a  factor  of  Z:,  22;  —  5  a  factor  of 

Jci,  and  22+  5  a  factor  of  Ic_i ;   hence,  taking  7?i  =  5, 

9  7 
we  are  directed  to  try  the  substitution  —  for  x. 
•^  5 


2z 


10,- 


-(15y+4.),- 
Az 


-  (40y2-6y2),  (60/+16y^;2),  -24y^2; 
-^yz  -l^y'^z       2^fz 


-Sf 


12/; 


0 


126  FACTOEING. 


Sence,  5x  —  2z  is  one  factor,  the  other  being 
2x'  -  ^yx'y  -  8^/  +  12/. 

Seeking  to  determine  the  factors  of  this,  we  obtain 

m  =  ±  1  or  ±  2,  ^'  ==  12,  l^    =  3,     h,    =  0, 
X^_i  =  15,  /j_2--0. 

The  vanishing  of  Jc2  shows  that  a;  — 2 y  is  a  factor;  and 
the  vanishing  of  Jc__2  shows  that  ^  +  2y  is  also  a  fac- 
tor. Dividing  these  out,  the  remaining  factor  is 
found  to  be  2ii;  — 3y;  so  that  the  proposed  poly- 
nome  resolves  into 

(5x-2z)(x-2  y)  {x-\-2y)  {2x~Zy), 

The  factor  ^x  —  2z  might  easily  have  been  got  by  the 
method  of  §  23,  but  the  present  solution  show^s  we  are  inde- 
pendent of  that  section.  It  may  also  be  obtained  by  re- 
arranging the  polynome  in  terms  of  y. 

Factor:  Ex.40. 

1.  2c(^-~2^x^^mx~2^',  x^  -1  x'^y  +  l(Sx2f  -I2f. 

2.  \2x'  +  bx''y  +  xy''  +  ?>y''',  Sx^-l4:X  +  (S. 

3.  ^x^  ~\bax  +  a?x~ba^\  2x^ +  ^x''y +1  xy'' ~-?>^. 

4.  2h'~1h^c-Wc'  +  hc'~4.c'- 

Iba^  +  A^  o?h  +  l?>ah''  -\2Jy\ 

6.  150:?;*  -  725 r>  +  931  x'^y''  +  920 xy^  -  llb2y\ 

7.  36 x'-  6(9  -7y)a;^-7(9  +Uy)x''y+  3(49  -40y)^/+180y'. 

8.  I0x^~-a^{lbij+^z)+x\^0y''+^yz)+x{m'i^-l^yh)~-2^''z. 

§  30.    If  the  polynome    a:r"+  Z>:r**~^+ -\-  hx  -{-  Jc,    in 

which  a,  h,  ,  h,  Jc  are  all  integral  and  n  greater  than  3, 

have  no  rational  linear  factors,  it  may  have  rational  quad- 


FACTORING. 


127 


ratio  factors.  Let  m  denote  a  positive  measure  of  a,  and  r 
denote  a  measure,  positive  or  negative,  of  Tc.  Tlie  rational 
quadratic  factors  of  the  polynome,  if  there  be  any,  must  be 
of  the  form  r)ix^  ~\-  qx  —  r.  To  determine  such  factors  we 
may  proceed  as  follows  : 

For  X  substitute  successively  three  or  more  consecutive 

terms  of  the  progression ,  3,  2,  1,  0,  — 1,  — 2,  — 3,  , 

and  denote  the  corresponding  values  of  the  p)olynome  by 

,  ^^3,  h^,  hi,  k,  Jc_i,  Jc_2,  Jc_s, Let  r^  denote  "a  measure 

of  y^s,  positive  or  negative";  rg  denote  *'a  measure  of  ki, 
positive  or  negative  "  ;  etc.  Then  mx'^  -{-  qx—7'  need  not  be 
tried  as  a  factor  of  the  polynome,  unless  an  arithmetical 
progression  with  q  as  common  difference  can  be  formed  from 

among  the  values  of ,  9m+r3,  4m+^2,  ^>^+^i,  ^',  ^>2'+r_i, 

4??i+'^-2,  977i+r_3,  ,  in  which  the  coefficients  of  vi  are 

the  squares  of  the  terms  of  the  series ,  3,  2,  1,  0,  -  -  1, 

-2,-3, 

Examples. 


18. 


h_ 


■  78. 


1.    Factor  :27*  — 3^'-  13^^'  + 36^-, 
m   =1,  k    =-~    18, 

ki    =  3,  7^2   ==  —     6,     Z^a    =  —   9, 

Ic^i  =  -63,     Ic^,=- 102,     /l^_3---81, 

Trying  for  rational  linear  factors  as  by  §  28,  it  will  be 
found  there  are  none.  We  therefore  proceed  to  seek 
for  rational  quadratic  factors.  To  do  this,  we  first 
tabulate  the  arithmetical  values  of  ^3,  rg,  7\,  

9m 
4m 


9 

1.     3, 

9, 

6 

1      '^ 

3, 

6, 

3 

1,     3, 

18 

1      2 

3, 

6, 

9, 

18, 

63 

1,     3, 

7, 

9, 

21, 

63, 

102 

1,     2, 

3, 

6, 

17, 

34, 

81 

1,     3, 

9, 

27, 

81, 

78 

1,     2, 

3, 

6, 

13, 

26, 

51,     102, 


39, 


0 

4  m 

9  m 
16  m 


m  =  1 


128 


FACTOEING. 


Taking  these  both  positive  and  negative,  we  next  tabu- 
late tlie  values  of  9m  +  r3,  4m  +  r2,  m  +  ri,  This 

done,  we  then  proceed  to  select  and  arrange  in  col- 
umns any  arithmetical  progressions  that  are  found  to 
run  completely  through  the  table,  one  term  of  the 
progression  in  each  line  of  the  table  in  regular  order, 
thus  : 


r;  — 


0, 

6, 

8, 

_9 

1, 

2, 

-2, 

0, 

2, 

-18, 

-9, 

-6, 

-^2, 

-20, 

-8, 

-98, 

-47,- 

-30, 

-72, 

-18, 

0, 

-62, 

-23,- 

-10, 

10,  12,  18, 

0, 

12 

3,    5,    6,    7,10, 

2 

7 

4, 

4, 

2 

-3,-2,-1,   1,    2,   3,   6,   9,18, 

6, 

-3 

-6,-2,    0,   2,   4,    8,10,22,64, 

8, 

-8 

-13,-2,    1,   2,   3,   5,   6,    7,10,21,38,55,106, 

10,- 

-13 

6,    8,  10,12,18,36,90, 

12,- 

-18 

3,  10,  13, 14, 15, 17, 18, 19, 22, 29, 42, 55,   94, 

14,- 

-23 

There  are  two  columns  of  progressions :  in  the  first,  r  =  6 
and  q  or  the  common  difference  is  2,  giving  the  trial 
factor  x^  -\-2x  --^]  in  the  second  column,  r=— 3  and 
5'  =  —  5,  thus  giving  the  trial  factor  x^  —  bx-\-?>.  On 
actual  trial,  it  will  be  found  these  are  the  factors  of 
x'  -  ^x^  -  13:r2  +  36a;  -  18. 


Factor  6^^  -  53  .r^  +  179:r^  -  299  2;'  +  260^  -  96. 

Here  m  may  be  1,  or  2,  or  3,  or  6. 

^  =  -96,     ^1    =-3,     Z:,=.4,     ^3  =  -9,     h  = 
Jc_,  -  _  893. 

The  factors  of  k,  Jci,  ,  are  : 


32 

9 

4 

3 

96 

893 

32, 


0    \J1.     A/,     A/1,     ,     cXiC    . 

2,     4,  8,  16,  32, 

I6m 

3,     9, 

9m 

2,     4, 

4m 

3, 

m 

2,     3,  4,     6,     8,  12,  16,  24,  32,  48,  96, 

0 

19,  47,  893, 

m 

FACTORING.  129 


I 


As  this  table  yields  no  complete  column  in  aritlimetical 
progression,  the  given  polynome  has  no  rational  linear 
factor.     (§  29.) 

Forming  the  table  for  vi=l,  it  will  be  found  that  it 
also  does  not  yield  any  trial  divisor. 

The  table  for  m  =  2  is  : 


0, 

16, 

24, 

28, 

30, 

31, 

33,  34,  36,  40,  48,  61, 

28 

9, 

15, 

17, 

19, 

22 

27. 

19 

4, 

6. 

7, 

9, 

lo! 

12, 

10 

-1, 

1, 

3, 

5, 

1 

-96, 

-48, 

-32, 

-24, 

-16. 

-12, 

-_8, -6, -4, -3, -2, -1,1,2,3, 

-8 

-891, 

-45, 

-17, 

1, 

3, 

21, 

49,  895, 

-17 

This  gives  the  complete  column  set  out  at  the  right. 
In  it,  r  =  —  8,  and  the  common  difference  is  —  9  ; 
hence,  we  have  2x'^  ■—9x-{-S  as  a  trial  divisor.  On 
actual  trial  it  will  be  found  to  be  a  factor  of  the  given 
polynome,  the  co-factor  being  3.r^  — 13:r'^  + 19^— 12. 

Ex.  41. 

Factor : 

1.  x'~12a^  +  4:7x'-66x  +  27, 

2.  x'~6x^-2x'  +  S6x~-24:. 

3.  x'-2x'-2bx'  +  lSx  +  24:. 

4.  :?;5-31^^  +  186r?;-180. 

5.  1  -  45a;^  +  S2x'  +  281  x'  -  518^^  +  252^«. 

6.  a'  -  38 a'f  +  2SaY  +  345 d'y'  -  664:ai/'  +  180/. 

7.  2:?;*- 5^^- 17:^'^  + 53^- 28. 

8.  6x'-  5Sx'  +  83a;*  +  45^;^  -  257^^  +  32:r  +  15. 

9.  6-47y  +  108y^-74y^  +  12/. 
10.  6:r^  -  17a;* +  5:^^^  + 13:^^-2^-2. 


CHAPTEK   IV. 

Measures  and  Multiples. 

§  31.  When  one  quantity  is  to  be  divided  by  another,  the 
quotient  can  often  be  readily  obtained  by  resolving  the 
divisor  or  dividend,  or  hoih,  into  factors. 

Examples. 

1.  Divide  a'  -  "lab  +  ^^  -  c^  +  'lcd~~-  d'' hj  a- b  +  c -d. 
Here  we  see  at  once  that  the  dividend 

=  {a-hy-{c~d)\ 

and  hence  quotient  —  a—b  —  (c~d)  =  a—b  —  c~\-d. 

2.  Divide  the  product  of  o?  -\-  ax-\-  x^  and  c^  +  x^ 

by  a*+a^^^  +  ^^ 
Here  o^  -\-  x^  ^=  {a -\-  x^  {o?  —  ax  -\-  x^),   and  the  divisor 

Hence,  the  quotient  i^  a-\-  x. 

3.  Divide  a^  +  a^b  +  o?c~  abc  —  b'^c  —  b&  by  d^  —  be. 
The  dividend  is  a  {c^  —  bc)-\-b  {c^  —  be)  +  e  (a^  ■—  be). 
Hence  the  quotient  ^=^  a-{-b  ~\-  e. 

4.  {a^  +  b^  —  c^J^^abe)-^{a  +  b-c). 

J)Wi^Qn^  =  a^  +  b^+?>ab{a+b)~e^~^ab{a  +  b)+^abG 
=  {a  +  by-c^~2>  ab  {a  +  b-c), 
which  is  exactly  divisible  by  a  +  ^  —  e. 

Quotient  =  a^  -{-  b"^  +  e"^  —  ab  -\-  be  +  ea. 


MEASURES    AND    MULTIPLES.  131 

5.  Divide  x^  —  x^y  +  x^y"-  —  29 if'  +  xy^  —  y^  by  x^  —  y^. 

The  dividend  is  (§  24)  evidently  {x^  —  y^)^{x-\-y^^ 
and  this  divided  by  x^  —  y^ 

=--  (^'  +  /)  -^  (^'  +  y)  =  ^''  —  ^y  +  y'- 

6.  Divide  h(^x^-\-c^)-^ax{x^—dF)-\-d'{x^a>)  by  (a+Z>)(.r+a). 

Striking  the  factor  x-\-  a  out  of  dividend  and  divisor, 
we  have  h  {pf  —  ax  -\-  o?)  +  ctx  (x  —  a)  +  a^ 
=  h{x'^  ~  ax  +  a^)  +  a  (x^  —  ax -\-  a^) 
=  (a  +  ^)  (x^  —  ax  +  a^). 

Hence,  quotient  =^  x^  —  ax-\-  al 

7.  Divide  apx^  +  x^  (aq  +  ^j9)  +  x'^  (ar  +  ^^7  -\-pc) 

+ :?;  (^^  +  br)  +  c?r  by  a:r^  -i-hx-\-  c. 

Factoring  the  dividend  (§  22),  we  have 

(ax'^ -{-bx-i- c)(px'^ -\-qx -{-r). 
Hence  the  quotient  equals  the  latter  factor. 

8.  Divide  6x' -  ISax'  +  ISa'x'  -  13a\x~5a' 

by  2x'^  —  Sax  —  a^. 

This  can  be  done  by  §  15.  The  divisor  is  2x'^ — a^  —  Sax, 
and  we  see  at  once  that  3^^  +  5a^  must  be  two  terms 
of  the  quotient. 

Multiplying  diagonally  into  the  first  two  terms  of  the 
divisor,  and  adding  the  products,  we  get  -i-7a^x^; 
but  +  13  a^^^  is  required.  Hence,  +  6a^x'^  is  still  re- 
quired, and  as  this  must  come  from  the  third  term 
multiplied  into  —Sax,  that  third  term  must  be  — 2  ax. 
Therefore,  the  quotient  is  3^^  +  5a^  —  2 ax. 

Note.  By  multiplying  the  terms  —2 ax,  —Sax,  diagonally  into 
the  x'^'s  and  a^'s,  respectively,  we  get  the  remaining  terms  of  the  divi- 
dend.   It  is,  of  course,  necessary  to  test  whether  the  division  is  exact. 


132  MEASURES    AND    MULTIPLES. 

9.  J)Wid.Q2a'-o?b~l2a^h''~bah^+Whjo?-~h''-2ah. 
Here,  as  before,  one  factor  is  a^  —  H^  —  2ah\  hence  two 
terms  of  tlie  otlier  factor  are  2o?  —  Ab'\  Multiply- 
ing, as  in  the  last  example,  we  get  — Qa'^h'^]  but 
—  12 a^ b'^  is  required.  Hence,  ~^a?b'^  is  still  needed, 
and  +3aZ>  is  the  third  term  of  the  required  quotient, 
which  is  therefore  '2ia^  —  4:h^-{-?>ab. 

10.    Prove  that 

(1  +  X  +  x'  + +  x''-^)  (l~x  +  x^~ +  x''-^) 

=  1  +  ^'  +  ^*  + +  :^'"-l 

Product 

^l—x""  ^l+x" 
■  X  ■ 


\~~  x      l-\-  X 
1  -  x"'' 


=  l  +  x'  +  x^+ +  a;2"-l 


11.  J)Wid.Q{a^  —  bcy  +  U^c^hjo?  +  bc. 

=:  (a'  -  bcf  +  (2  bcf  -^  (a'  -be) +  2  be 
=  (a^-  bey  -  {a?  -  be)  X2be  +  (2bey 
=  a'-4:a''be+7b'e\ 

12.  Divide  1  +  2,357,947,691 2;^  by  1  -  11:?;  +  121  ^^ 
Dividend    ==1  +  (11^)' 

-  [1  -  (11^)^  +  (11^)''][1  +  (11^)']. 
Divisor       =  [1  +  (11  rr)']  -^  (1  +  11:^). 
.-.quotient- [1- (11:^7 +  (11^)^(1 +  11^). 

Ex.  42. 

Find  the  quotients  in  the  following  cases : 

1.  1  — x-\-x'^  —  x^ -i-1  — X. 

2.  l-2x'  +  x^-^x' +  2x^  +  1. 


MEASURES    AND    MULTIPLES.  133 

5.  I~4:x''+l2x^-^x'^l  +  2x-?>x\ 

6.  (a'  -  2ax  +  x'')  (a''^  +  ?>a'x  +  Zax''  +  x"")  -^  a'  -  x\ 

7.  x^  —  y^  +  2;^  +  3 :r?/2;  -^  :i;  —  y  +  2;. 

8.  6a'~^a'b  +  2a'b'  +  ISab'  +  4:b'-^2 a'  ~  3ab  +  Ab\ 

9.  4:x'~x^f  +  6x7/  —  9f-i-2x'  +  Sf~xy. 

10.  a'  +  b'-c'-2a'b'---a'-b'~c\ 

11.  21a'^-16a-^6  +  16a'Z)'-5aZ>^  +  2Z»*^3c^*^-a5  +  5l 

12.  2a^-7a'-46a-21--2a'  +  7a  +  3. 

13.  [a'(b  -c)  +  P(c  -a)  +  c'(a-h)]  ~a  +  b  +  c. 

14.  x^ —  dax'^  +  Sa^x  — a^-{-b^~7-x~a  +  b. 

15.  x'  -y'  +  z'  +  2x''z'  —  2y'  -  1  -^ x' ^f  +  z''-l, 

16.  :?;*  —  (a  +  c)  x^  +  (^  +  ac)  :r^  —  bcx  -^  x  —  c. 

17.  r'^  +  ^'y  +  :r?/'  +  ?/-'^--.'^  +  y. 

18.  x''  —  ^^y  +  rr^y'  -  x'y^  +  x^7/~-x''y'^  +  xy^-~y'^~^x^-{-7/\ 

19.  a*  +  Z)*  —  6'^^  -  2a' 5'^  _  2c'  -  1  --  a'  ~  b'~c'  -  1. 

20.  a'  -  ab^  -  ac^  -  2a'^  +  2Z>^  +  2bc'  +  ?>a^c-?>¥c-?>c' 

-^a  +  ?>c  —  2b. 

21.  a'Z>  —  ^a;'  +  a'^  — r?;^-^  (:r  +  ^)(a  — :r). 

22 .  a{b  ~  cf  +  b{c  —  aj  +  c{a  -by  -^  o?  -  ah  -ac  +  be. 

23.  a'  5'  +  2 a5c'  -  a' c"  ~  b'' d" -^  ab  +  ac  -  be. 

24.  ^•'+y^+3:?;y~l^:r  +  y-l. 

25.  x^  —  x^  —  2-^x''  —  x+l. 


134  MEASURES    AND    MULTIPLES. 

26.  a*  — 29a'  — 50a— 21 -^- a'  — 5a— 7. 

27.  (2x  —  yjof'  —  (^  +  y)'a':r'  +  2(^2;  +  y)ax^  —  x^ 

-^  {2x  —  y)  a^  -]-  {x -\- y)  ax  —  x^. 

28.  (^^-l)a^-(:u'  +  ^^'-2)a'+(4^'+3:i'+2)a-30r+l) 

-^(:?;-l)a'-(:2;-l)a  +  3. 

§  32,  The  Highest  Oommon  Pactor  of  two  algebraic  quan- 
tities may,  in  general,  be  readily  found  by  factoring.  The 
H.C.F.  is  often  discovered  by  taking  the  sum  or  the  differ- 
ence (or  sum  and  difference)  of  the  given  expressions,  or  of 
some  multiples  of  them. 

Examples. 

1.  Find  the  H.C.F.  of  (h  —  c)x'+  (2ah -2ac)x  +  a'b  -  a'c 

and  (ab  —  ac  -{-  h"^  —  he)  x  -{-  a"^  c  -\-  alf  —  a^h  —  ahc. 
Taking  out  the  common  factor  h  —  c,  we  get 

(h  —  c){x'^+2ax-\-ah)  and  {h  —  c)\{a—h) x  —  a^+ah'\. 
Therefore,  h  —  c\^  the  H.C.F.  of  the  given  expressions. 

2.  Find  the  H.C.F.  of  1  — x  +  y +  z  — xy  +  yz  —  zx  —  xyz 

and  \  —  x  —  y  —  z  +  xy-\-yz-\-zx  —  xyz. 
Their  difference  is 

2y  +  2z  —  2xy—  2zx  =  2(l  —  x){y  +  z). 
Their  sum  is 

2  —  2:r  +  2yz  —  2xyz  =  2  (1  —  x)  (1  +  yz). 
Therefore,  the  H.C.F.  is  (1  -  x). 

3.  Find  the  H.C.F.  oix>  +  ?>x'-  8;r'  -  9a;  -  3 

and  a^  -  2x'  -  e>x^  +  4:r^  +  13:r  +  6. 

The  annexed  method  of  finding  the  H.C.F.  depends  on 
the  principle  that,  if  a  quantity  measures  two  other  quanti- 
ties, it  will  measure  any  multiple  of  their  sum  or  difference. 


MEASURES    AND    MULTIPLES.  135 


1 
1 

+  3 

—  2 

0 
-6 

-8 
+  4 

-9 

+  13 

-3 
+  6 

(«) 
(b) 

(c)  [= 
(d) 

if) 

-45 
-45 

5 

+  6 

-12 

—  22 

-9 

-(a)~(b)] 

2 
1 

+  6 

-2 

0 

-6 

-16 

+  4 

-.18 
+  13 

-6 
+  6 

(a)  X  2 
(b) 

3 

+  4 

-6 

-12 

-5 

-27 
-25 

-2 

' 

15 
15 

C.F.= 

+  18 
+  20 

—  2 

-36 
-30 

-6 

-66 
-60 

-6 

(.)X3 
(£^)X5 

1 

25 

27 

+  3 

+  30 
+  36 

+  3 

-60 

-54 

+  1 

-110 

-108 

(e)x5 
(c;)x9 

2 

-6 

-6 

-2 

(9) 

H. 

1  +3 

^{x  +  iy. 

+  3 

+  1 

The  coefficients  are  written  in  two  lines,  (a)  and  (h). 
They  are  then  subtracted  so  as  to  cancel  the  first 
terms,  (a)  is  next  multiplied  by  2,  and  added  to 
cancel  the  last  terms.  If  (c)  and  (d)  had  been  the 
same,  their  terms  would  have  been  the  coefficients 
of  the  H.C.F.  Since  they  are  not,  we  proceed  with 
them  as  with  (a)  and  (h)  till  they  become  the  same. 
When  (a)  >nd  (b)  do  not  contain  the  same  number 
of  terms,  it  is  more  convenient  to  find  only  (c),  and 
then  use  this  with  the  quantity  containing  the  same 
number  of  terms.  The  general  rule  is  to  operate  on 
lines  containing  the  same,  or  nearly  the  same,  num- 
ber of  terms. 


136  MEASURES    AND    MULTIPLES. 

4.    Find  the  H.O.F.  of 

3  a;'  +  2  :^'  -  14  ^  +  8  and  6  ^'  -  1 1  ^'^  +  13  ^  -  12. 


8      +2    -14     +8 
6    -11    +13    -12 

(«) 
id) 

6     +4-28+16 

(a)x2 

15    -41    +28 
(5-7)(3-4) 
H.C.F.  =  3a;-4. 

(b)~(a) 

If  (a)  and  (h)  have  a  common  factor,  its  first  term  must 
measure  3  and  6,  and  its  last  term  must  measure  8 
and  12.  (<?)  is  not,  therefore,  the  H.C.F.  Eesolve 
(c)  into  factors,  bx—7  is  not  a  factor  of  (a)  and  (b). 
If,  therefore,  (a)  and  (b)  have  a  common  factor,  it  is 
3^  —  4.  On  trial,  3^  —  4  is  found  to  be  a  factor  of 
(a),  and,  therefore,  it  is  the  H.C.F.  of  (a)  and  (b). 

,    If  x"^  -{-px  +  q  and  x'^  +  r:^  +  5  have  a  common  factor, 

prove  that  this  factor  is 

q  —  s 

p  —  r 

If  x~ahe  the  common  factor,  then  the  remainders,  on 

dividing  the  given  expressions  hj  x  —  a,  must  be  zero ; 

that  is,  a^  ~{-pa-\~  q  =  0, 

and  a^  +  ra  +  s  ===  0,  or  (p  —  r)  a  =  s  —  q. 

s  —  q^  s  —  q  q~s 

Hence,  a  = ,  and  x  —  a=x —  x-] 

'         p~r  p  —  r  ^  p  — r 

,    What  value  of  a  will   make  a^x"^ -{- (a  +  2)x -{-1,  and 

a^ x"^  -{-  a^  —  ^  have  a  common  measure. 
They  cannot  have  a  monomial  factor.     Neither  can  they 
have  one  of  two  dimensions  unless  (a  +  2)  vanishes  ; 
that  is,  unless  a  —  —  2,  in  which  case  the  expressions 
become  4:r^  +  1  and  4:X^  —  1,  which  have  no  common 


MEASURES    AND    MULTIPLES.  137 

factor.  Hence,  if  the  given  quantities  have  a  com- 
mon factor,  it  must  be  of  the  form  x  -{-m]  dividing 
a^ x"^ -{-  a^  —  5  by  x  +  m,  we  have  for  remainder 

5  —  a'^  1 

a^m^  +  a^  — 5  =  0  or  m^= — ;  .'.  m  =  - -y/(5 —  a^), 

in  which  -^(5  —  a^)  must  be  possible  and  integral ; 
hence,  a^  =  4:  (a^  =1  gives  values  to  7n  which  on 
trial  fail)  and  a  =  ±2,  of  which  the  positive  value 
must  be  taken;  and,  therefore,  2x  +  1  is  the  common 
factor. 

If  the  H.C.F.  of  a  and  b  be  c,  the  L.C.M.  of 

(a  +  h)  {a?  -  ¥)  and  (a  -  h)  (a'  +  F)  is  ^J^'. 

Let  a  =  mc,  b  =  nc,  and  .*.  a^  =  w? &,  b^  ~  n'&. 
Thus  (a  +  Z*)  =  c{in  +  n) ;  (a  —  b)  ^=  c (m  —  n), 

and  (a^  +  b^)  =  &  (m^  +  n^)  ;  (a^  —  b^)  —  &  {rn?  —  72^). 
Hence,  {a  +  Z>)  (a*'^  —  b^^  —  c^  (in  +  7i)  (m^  —  7i^), 

and    (a  —  5)  (a^  +  b^)  ^  c^{m~  n)  (m^  +  n^). 
The  H.C.F.  of  the  last  expressions  is  c*(m^  —  n^)  ;  hence, 

the  L.C.M.  =  cHm'-n')^^^;^^'^^^' 

I^  (x  —  ay  measures  x^  +  qx  +  ?-,  find  the  relation  be- 
tween q  and  r. 

Let :?;  +  m  be  the  other  factor  ;  then 
x^  -}-  qx  -i- r  =  (x  —  ay(x  +  m) 
^=x^  -{-  (m  —  2d) x'^  -{-  (a?  —  2  am) x  +  nid^. 

Equating  coefficients,  m— 2a  ==0,  a^—2am--=q,ma^  —  r. 

r         ft      r"^ 

or  a  =  — • 

4 


Hence,  m 

=  2a, 

and 

•.  «^ 

-4a^ 

=  ^,  2a' 

and  a^  = 

_      9 
8' 

or  <:/ 

=  -- 

1^  ;  and  a'  = 

Therefore, 

4~~ 

27' 

^27 

=  0. 

138  MEASUEES    AND    MULTIPLES. 

Or  thus  :  Dividing  x^  ^  qx -\~  r  by  (x  —  a)'^  we  find  the 
remainder  (q  -\-  S a^)  x  -\-  r  —  2a^,  and  as  this  will  be 
the  same  for  all  values  of  x,  we  have,  by  equating 
coefficients, 

^  +  3  a^  ^  0,  and  r  -  2a'  =  0, 
or  q^  —  27 a^  and  r^  =  4:a^ ; 

therefore  —  +  ^  =  0,  as  before. 

Ex.  43. 

Find  the  H.C.F.  of  the  following: 

1.  2x'  +  3x'+bx'  +  9x~S',  Sx'-2x'+10x'-6x  +  d. 

2.  x'+(a  +  l)x''+(a+l)x  +  a]  x'+(a-l)x''-(a-l)x+a. 

3.  px^—(p—q)x'^-\-(p—q)x-{-q ;  px^—(p-\-q)x'^-\-(p+q)x—q. 

4.  ax^—(a-h)x^-(b-c)x-c]  2ax^+(a+2b)x''+{b+2c)x+c. 

5.  l-^^x-S^x^+^x'-x']  l-lji^x~3x'  +  lj\x'+x\ 

6.  ac'«  +  ^c?'*  +  (a  +  5)c«+^  a'c'' +  a'c' +  c^'b' +  b'c\ 

7 ..   a V  +  a^-2  abx"  +  Z>  V  +  a^b^  -  2  a'b 

and  2  aV  -  5  aV  +  3  a«  -  2  Z> V  +  5  a'bV  -  3  a'b\ 

8 .  (ao;  +  byy  ~{a  —  b)  (x  +  2;)  (ax  +  Z^y)  +  (a  —  Z>)^  ^2; 

and  {ax—byf  —  (a  +  5)(ri;  +  z){ax  —  Sy)  +  (a  +  hfxz. 

9.  a(Z>^-c^)  +  Z>(d^-aO  +  c(a^-&0 

and  a  {ly^  -&)-\-b  {c'  -  a')  +  ^  («'  -  ^0- 

10.  a     +  a'"*  +  6^  +  1  and  a'^  -  a'"*  +  a^'^  —  1. 

11.  \i  X?  ^r  (^^^  -\-bx-\-  c  and  x^  +  a':?;  +  Z*'  have  a  common 

factor  of  one  dimension  in  x,  it  must  be  one  of  the 
factors  of  (<x  —  a')  x'^  -{- (b -~b^)x  +  c. 

12.  Determine  the  H.C.F.  of  (a  -  Z>)^  +  (b  -  cf  +  (0-  a)' 

and  (a'  -  bj  +  (b'  ~  cj  +  (c'  -  d^. 


MEASUEES    AND    MULTIPLES.  139 

13.  Find  the  H.O.  F.  of 

and  3(y'  -  4y'  +  5y  -  2)x' 

14.  If  x^  -\-px  +  q  and  x"^  +  '^^  +  "^  have  a  common  linear 

factor,  show  that 

(n  —  qy  +  '^{'^  ~~ pT  =  7n{m — p)(n  —  q). 

15.  Find  the  L.O.M.  of  :r'- 3.^2+3:^-1,   x^-c^  —  x+l, 

x'-2o(?  +  2x-l,  2.ndix'~2x'  +  2x'  —  2x+l. 

16.  Find  the  L.C.M.  of 

x^-\-^x''  +  llx+Q,   x''  +  1x''  +  Ux  +  ^, 

x'  +  Sx"-  +  19^  +  12,  and  x'  +  dx'  +  2Qx  +  24. 

17.  Find  the  value  of  y  which  will  make 

2(y'  +  3/)^'  +  (lly  -  2):^  +  4  and 

2(f  +  y')x'  +  {ll\/--2y)x'  +  (f  +  5y)x+5y-l 

have  a  common  measure. 

18.  The  product  of  the  H.C.F.  and  L.C.M  of  two  quanti- 

ties is  equal  to  half  the  sum  of  their  squares  ;  one  of 
them  is  2x^  —  llri;^  +  17:^:  —  6  ;  find  the  other. 

19.  If  :^  +  a  and  x  —  a  are  both  measures  of 

x^  -\-px'^  -{-  qx  -[~T,  show  that  pq  —  r. 

20.  li  x^  -\-  qx  -}- r  and  x^  +  mx  +  n  have  a  common  meas- 

ure (x —  ay,  show  that  q'^n^  =  rn?r^. 

21 .  If  the  H.C.F.  of  x^  -{-px  +  q  and  x^  +  mx  -\-nhQ  x-\-a, 

their  L.C.M.  is 

x^-\-  (m  —  a)  oi^+px^-^  (a^+  mp)  x-\-a(m  —  d)  (a^+p) . 

22.  li  x^  -\~  qx+1  and  x^  -\-px^  +  5'rr  +  1  have  a  common 

factor  of  the  form  x-^-a,  show  that 
(p-l)-'-^(p-l)  +  l  =  0. 


140  FRACTIONS. 


23.  If  x^ -j- px^ -}~  q  smd  x'^-\-mx-\-n  have  x -\- a  for  tlieir 

H.O.F.,  show  that  their  L.C.M.  is 
rr*+  (m  —  a  +p) 0(^  +p (jri  —  o^x"^  -\-  c^ {a—p)x 
+  a?  (a  —p)  (m  —  a). 

24.  If  x'^  +px  +  1  and  a^  -\-px^  -^  qx-^X  have  x  —  a  for  a 

common  factor,  show  that  a  = 

\-q 

25.  Find   the   H.C.F.    of  {a^  -V^J  -\-{})'  -  &y  ^{c"-  -  a^J 

2.Tidd'{h  —  c)-\-l\c~d)Arc\a~h), 

26.  If  a  be  the  H.  C.F.  of  I  and  c,  p  the  H.C.F.  of  c  and  a, 

y  the  H.C.F.  of  a  and  ^>,  and  S  the  H.C.F.  of  a,  ^>, 

and  d?,  then  the  L.C.M.  of  a,  Z>,  and  c  is 

27.  If  ^+^  be  the  H.C.F.  of  the  x'+ax+h  and  x''+a^x+h\ 

their  L.C.M.  will  be 

x^ -\-{a  -\-  o!  —  c)x^  +  {aa^  —  &^x-\-  {a  —  c){o}  —  c)c. 

28.  Show  that  the  L.C.M.  of  the  quantities  in  Exam.  2 

(solved  above)  will  be  a  complete  square  if 
X  =^  y^  -\- z^  — 'if  z^. 

29.  Find   the    H.C.F.    of  x^  +  2x^  +  3^*  -  2x''  +  1  and 

^x^  +  x' +  11  x'' -1x^-2. 


Fractions. 

§  33.  When  required  to  reduce  a  fraction  to  its  lowect 
terms,  we  can  often  apply  some  of  the  preceding  methods 
of  factoring  to  discover  the  H.C.F.  of  the  numerator  and 
denominator. 

Examples. 

1      ac  +  hy  +  ay  +  he   __  c{a  +  h)-\-y(a+h)  __  c  +  y 
af+2bx  +  2ax+bf  ~ f{a+h)  +  2x{a+b)~ f+2x 


FRACTIONS.  141 


a^  _  ha'  -  a'h''  +  ah\_^a\a'  +  h'~ah{a  +  b)'\ 
a'  -  ba'  -  a¥  +  h'   ~  a{a' -  h')  -h{a'-  h') 

_  g  (g  +  h)  {a  —  hy_       a 

~{a-h){a'-h')  "  a^  +  h"        ■ 

x^  —  x^y-\'0(?  y^  —  x^  y^  +  xy^  —  y" 

Here  the  numerator  is  evidently  (x^  —  ?/)  -^  (:r  —  y),  and 
the  denominator  is 

• ^.     The  result  is  therefore ^• 

{x-\-yy'  —  x'  —  ^_    hx'^y  +  lOrg'y'  +  Wi x^ y^ -\- h xy^ 
{x  +  yy-\-x'^y'~  {x^yy-x'y''  +  {x?  +  y''y-x'f 

_^ hxy\x?  -\-f-\-  2xy  {x  +  y)] 

(^'  +  y'  +  ocy)  [{x  +  yf  +  xy  +  x^  +  y''-  xy] 
bxy{x  +  y)  {x^  +  xy+y'')  _     bxy{x  +  y) 
^{x'  +  xy  +  yy  2{x'  +  xy  +  yi 

x''~\2x  +  ?>b 


5. 


x^~\Ox'  +  ?>lx  —  ^0 

Here  we  see  at  once  that  the  numerator  =^{x~b)(x  —  l) ; 
and  it  is  plain  that :?:  —  7  is  not  a  factor  of  the  denomi- 
nator ;  we  therefore  try  x—b  (Horner's  division),  and 
find  the  quotient  to  be  o;^  —  5  ;r  +  6. 

Hence,  the  result  =  — — -• 

X   ~DX-\-0 

x'^2x^  +  ^ 


x'  —  ^x'+Sx-^l 

The  factors  of  the  numerator  are  at  once  seen  to  be 
x^  -\-2x-{-?>  and  ^^  —  2  a;  +  3,  of  which  the  latter  is 
one  factor  of  the  denominator,  the  other  being  (Hor- 
ner's division)  x^  —  2x —  7. 

Hence,  the  result  is  —-^ ^t_^. 

x'-2x-l 


142  FRACTIONS. 


Ex,  44. 

Reduce  the  following  to  their  lowest  terms : 

1  ^^  ~~  "^  ^  +  ^        '     3  xy~  —  13  rry  -f  14  ^ 

x'-2x'-Sx-m'        73/^-173/^  +  6?/ 

„  x^-\~  aV  +  a'^      .  x^  -\-x~12 

x^  +  ax^  —  a^x  —  a^      x^  —  bx"^  -{-7  x —3 


a^-3x  +  2.  x'  +  2x''  +  9 


x^  +  4:x'  -  5       x^-  4.r'  +  Ax'  —  9 

2  +  bx .     x'  +  2x''  +  2x 

2b  +  (b'-A)x~2bx''  x'  +  4:x 


a^x  +  2aV  +  2  ax^  +  ^*       25^*  +  5^^  —  :r  —  1 

rz:*^  —  x^y  +  ^^y'^  —  ^r^  +  a^y^  —  ^^y^  +  xy^  —  3/"^ 
ri;^  +  x^y  +  :i;^3/'^  +  ^y  +  x^y^  +  rr''^3/^  +  xy^  +  3/^ 

3aV~2aa;^-l  y^      a)  ^  ^ "^ 

•    4aV-2aV-3a^^  +  l'     ^.  + g  __.^Yr3/-3/^' 

g     d'(b  -c)  +  b\c ~a)-\-c\a~  b) 
abc  (a  —  b)(b  —  c){c  —  a) 

<,     {a  +  h  +  cr . 


a\b  -  c)  +  b\c~a)  +  c'(a  -  b) 

10.  From  Exam.  4  (solved  above)  show  that 

{a-bY+{b~cy+{c-aY  _  {a-by+{b-cf+{c-af 
{a-bf+{b-cy+{c-af        b{a-b){b-c){c-a) 

11.  '(^  +  y)'-^-y'. 

{x  +  yf -x^-y'^ 


FRACTIONS.  143 


12.    Show  that 

{a~hy  +  {h  -  cy  +  {c  ~  ay 

(ci-hf  +  {h  -  cf  +  {g  ~  af 

=  i^[{o.-by  +  (h-cy  +  {c-ayi 

§  34.  In  reducing  complex  fractions  it  is  often  convenient 
to  multiply  both  terms  of  the  complex  fraction  by  the 
L.C.M.  of  all  the  denominators  involved. 

Examples. 

1.    Simplify  ^^--  +  ^^)--^^---X 

Here  the  L.C.M.  of  all  the  denominators  involved  is 
12 ;  hence,  multiplying  both  terms  of  the  complex 
fraction  by  12,  and  removing  brackets,  we  have 
6a:  +  8  — 8  +  6^;^    12^;    _    Zx 
21-4rr-17         4-4.T      \-x 

a  —  b 


a  ~  -_ 


l  +  ah 


-J    ,  a{a~b) 


1  +  ab 

Here,  multiplying  both  terms  by  l-\-ab,  we  get 
a{l  +  ab')-a  +  b  _b {a'  +  1)  _  ^ 
l  +  ab  +  a{a-b)        a' -f  1 

1 


1  + 


4  —  ^ 

Here,  multiplying  both  terms  of  the  fraction  which 
follows  :?;  —  1  by  4  —  a;,  the  given  fraction  becomes 

1 

at  once ; 

1    I  4  —  ^ 


144  FRACTIONS. 


and  now,  multiplying  both  terms  by  4,  we  have 

4  _    4: 

4^  —  4  +  4  —  ^      3x 
It  may  be  observed  that  when  the  fraction  is  reduced 

to  the  form  7-^3,  we  may  strike  out  any  factor  com- 
mon to  the  two  denominators,  and  also  any  factor 
common  to  the  two  numerators ;  it  is  sometimes  more 
convenient  to  do  this  than  to  multiply  directly  by 
.    the  L.C.M.  of  all  the  denominators. 

A     a-      T-r     fa  +  h  ,  a  —  h\      fa^  +  h^      a^  —  h' 
4.    Simplify  '       ^ ' » -^  ' ^ 


a-b  a  +  hj  \a'-~b'  a^ -\-h\ 
Here  the  numerator  of  the  first  fraction  is  {a  +  by  + 
{ct  —  b)\  and  the  denominator  is  a^  —  b'^\  the  numera-: 
tor  of  second  fraction  is  (a^-\-b'^y  —  (o^  —  b'^y,  and  the 
denominator  is  a^  —  b^\  the  former  denominator  can- 
cels this  to  a^  +  5^  which,  of  course,  becomes  a  multi- 
plier of  the  first  numerator. 

Hence,  we  have  ^^'Y?^tt'Jt^V^^^  =  ^4±ff 

Occasionally,  we  at  once  discover  a  common  complex 

factor ;  strike  this  out,  and  simplify  the  result. 

5.    — ^ ^ Here  the  denominator 

a^      y^      c^      ab 

a      b)      c^      \a      b      cj\a      b      c 
and  cancelling  the  common  factor,  we  have 

,  and,  multiplying  by  abc^  this 

a  '  b      c 
_  abc 

be  -\-  ca—  ab 


FKACTIONS.  145 


I 


Ex.  45. 

Simplify  the  following : 

a+b  .  a~h 

l_i[l_|(l_^)]'     gj^b      a~b 
a—b      a+b 

XX  _1^ 1_ 

^     x^y      x  —  y       \~a      1  +  a 
/&,  ' ^  j  I 

2.T  a      ,       1 


x^  —  if'  1  —  a      1  +  a 

1 


1  + 


1  +  a 


1  +  a 


2a^         a^  +  ^\       a      ab^ 
4  ' 


2b''        d'  +  b''  ~^b 

a-\-b  .  a~-b  ^   j    ,    b'' 

—7—,  +  — :j  a  +  b  +  - 

^     c  -\-  a      c  —  a  ,  a 

o.    ,  . 

a-Vb  ,  a  —  b  ,   j    .    a' 

_IE^  +  _^       a+b+  — 

c  —  a      c  -\-  a  b 

x—1  ,  y  —  1  ,  2  —  1 

Sxt/z  X  y  z 

'    yz  —  zx  —  xy  -'-   i   -^   i  -'- 

X      y  z 

2       2       2      d  +  ¥-\-c' 

^    d     b-"     c'       dh'c' 

a  A    b        c 
be      ac      ab 


146  FRACTIONS. 


o. 


a  —  b      o?  —  hy      \a  +  b      a^  -{-h' 
10.    <^__b+J_(i  ^^+1 


1  IV  2.bc 


11. 


12. 


13. 


14. 


a      b  +  c 

2(1- x)      (1  -  xy      ^       /x  -  aV     /a;  +  a 

l-x    ^\l-xj^ 


x^-aV  fx+a 

x  +  aj'^^'^ 


o  /a  +  ^\  __  o  fa  +  ^V"^  I   fa  +  b 


x^  ~  x^  y -\- x^  y^  —  x^  1^ -\- xy^ -- If'  _   fx  —  y 
x^  +  x^x  +  x^y"^ +  x^if  +  xy^  +  y^  '   \x  +  y^ 


16.  Find  the  value  of 

— — 1 y wlien  X  =  }  (a  +  b). 

2na  —  2nx      2nb  —  2nx 

17.  Find  the  value  of  V[l  -  V(l  ~  ^)] 


FRACTIONS.  147 


18.    Find  the  value  of 

^(a  +  hx)+^{a-hx)  ^^^^  ^  _      2ac 
V(«  +  bx)  -  V(«  -  bx)  ^  (1  +  c") 

§  35.  When  the  sum  of  several  fractions  is  to  be  found, 
it  is  generally  best,  instead  of  reducing  at  once  all  the 
fractions  to  a  common  denominator,  to  take  two  (or  more) 
of  them  together,  and  combine  the  results. 

Examples. 
1.    Find  the  sum  of 

x~\-y  y  —  X    _  o^j—j^ 

2x-2y      2x  +  2y     x'  +  y''' 

Here,  taking  the  first  two  together,  we  have 

{x  +  yy  +  {x~  yy  ^x"  +  f  . 

^{x'-f)  x'-y''' 


now  add  this  to  — 


x'  —  y^ 


x'  +  y' 

.  ^  (x'  +  yy  -  (x'  ~  yj       4  x'  y' 

and  we  get  ^     ^ '^  \ — ^ ^^^  ^  -^-^^ 

x^  —  y*  x^  —  y 

2.    Find  the  sum  of 

1  +  ^  .      AiX      ■      8;r        \  —  X 


\~x      \-\-x^      l-Yx'      \  +  x 
Here,  taking  the  first  and  the  last  together,  we  have 
(1  +  xy  -  (1  -  xj  ^     ^x     . 
1-x'  l-x'^ 

taking  this  result  with  the  second  fraction,  we  have 


,1  +  x''      l  —  xy      1  — a;* 
now  take  this  result  with  the  remaining  fraction,  and 
we  get 

1       ,       1     \        16^ 


l  +  x'J      1-a^ 


148  FHACTIONS. 


x"'  X''^  1         ,  1 

O.       1 ; — —  — -j-  - 


1        X''+l        ^^—1        ^^  +  1 

Taking  in  pairs  those  whose   denominators  are  alike, 
we  have 

^3n  .  1  ^2n 1 


^"  ^  1  X^'+l 

The  work  is  often  made  easier  by  completing  the  divi- 
sions represented  by  the  fractions. 

4.    Findthesumofl+-^^±i~l^^- 

2(07- 1)      2^  +  2 

By  dividing  numerators  by  denominators,  this 


2:r-2  2^  +  2      2.r-2      2:i;  +  2 


X       ,  X  —  9      X  -{-  1      x  —  8 

5. 1 * 

X  —  2      X  —  7      X  —  1      X  —6 


We  have,  by  division, 


2      .1-     2     _i. 


X  - 


2  x~7  x—1  x—6 


2  2  2  2 


:?;  —  6      ^  —  7      a;--l 

_     2(2^-8)  2(2.T-8) 

(x-2)(x-6)      (x~l)(x~7) 

:^ax-m( ^ ^ 

"^  -^^^-8^  +  12     x'-8x+7j 

=  (80  -  20 :r)  -^  (x'  -  16x'  +  S3x'-  152:?;  +  84). 
[Denominator  =  (x''  -  Sxf  +  19  (x''  ~8x)  +  84.] 


FRACTIONS.  149 


6.    Find  the  value  of  ^+|^  +  ^^  when  x  =  ^. 


By  division, 


4«  .  +  l  +  _i^_2  +  4f-^^+- 


x  —  2a  x  —  2b  \X'—2a     x—2b^ 

but  the  quantity  in  the  brackets 

^j^+P^^zA^      0,  since  {a  +  b)x  =  ^ab. 
(x-2a)(x~2b)  y    ^   J 

Hence,  the  value  of  the  given  expression  is  2. 

Ex.  46. 

Simplify  the  following : 

x  —  a  ■  cc^^ax  +  a^      x^  —  a^ 
5  x-\-  a  x^  —  a^ 

2  a^  +  b^  a'~^a^b  +  ^ab''-b^      a(a-b)-b(a~b) 
a'-ab  +  b'  a'-b'  a'+ab  +  b' 

3  /    1      .      1      .      2a    \/    1  1  2x 


\a-\-x     a  — 


4.    ^^  + 


:?;      a^  ~\-x'^J  \a-\-x      a~x     cr  -\-  x' 
b  ab       ,       ab 


5. 


a  +  b      a  —  b      ab  —  V^      d^  +  ab 

3  +  2^  _  2 -3a;  ,  l^x-x^ 
2-x        2  +  x         x'-4:  ' 


e.  :r..^.+        ' 


^     lfSx  +  2y\      lfSx~2y 
■    2\3x-2yJ      2>^3^+2y, 

8X —\~  ^           «^     ^ -L              -L       O^        I  it/               , 

•    7^ 7~7; — TT 7^ ^7-T+      /.    9 ^TT  + 


9. 


2a;-l      2a;+l      a;(l-2a;)      a;(4a;^-l)      :^(16a;*-l) 

1 4  9  x-1 

2a;  +  2      ^•  +  2"^2(.r  +  3)      (x  +  2)(x  +  3J 


150 


FRACTIONS. 


^{x  +  y)      2{y-x)      ^{x'-f)  ,  ^{x'  +  y') 


10. 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


22. 


x  —  y  x  +  y 


^  +  f 


■r 


)'  {x+^y] 


-^2 


X -{-a      x-\-  h^ 


fa  +  X        4:  ax         Sd^x       a  —  x\ 
\a~x      d^-j-x"^      a^-\-x^      a  +  xj 

■  \d'-x^~^  a'  +  x^     d'  +  xy' 

5x-4:.12x+2_  10:r  + 17 


9  11^-8 

a       ,       a       , 


18 

a' 


2  a' 


■ab' 


a'  +  b'      a'-b'      (a-b)(d'  +  b') 
12 :r  +  10a ^  117a  +  28^  _  ^g 


a'-b' 


3x-\-a 


9a  +  2x 


-3      5x-A 


4:x-17      8:t'-30  .  10:^- 

x-4:         2a;-72:?7-5        x-l' 

-n-    -I  XT,        1         n  a  +  b  +  2c  ,  a  +  b  +  2d 

Find  the  value  of       '       '        -| '       '       , 

a  +  b  —  2c      a-\-b~-2d 

when  a-\-b- 


c  +  d 


a^"" 


yn^^ 


r 


f 


^n_yn     x^  +  y'^     x^'—^f     x^'  +  y'' 
{a-by^  {a~by^  1 


^a-bf-l      {a-bf+1      {a-by-l      {a-by+l 

__1 + I I 

{a'-h'){x'+b')  ^  (b'-a%x'+a')      {a^+a%x'+b') 

l+x   ,    l-x  _     2     _2^ 
l_^3-r^_^^3      ]L-a;^      x^  +  \ 

o?+o?b  +  ab'+b'      (a+by-Sab      (g-bf-a'+b' 
a'-d'b-ab'  +  b'      (a-bf+Sab      (a  +  bf-a'-b' 


FRACTIONS.  151 


§  36.  The  following  are  additional  examples  in  which  a 
knowledge  of  factoring  and  of  the  principle  of  symmetry  is 
of  advantage. 

Examples. 

'    {x  +  zy~^f~^{y  +  xf-z^^{z  +  yy-x' 

Cancelling   the   common  factor  x  —  y  -{-  z  in  the  two 

terms  of  the  first  fraction,  there  results  ~ ; 

x  +  y  +  z 

hence,  by  symmetry,  the  denominators  of  the  other 

two  fractions  will  be  x-{-y  -\-z,  and  the  numerators 

will  be  y-\-z~x,z-\-x  —  y.     Hence,  the  sum  of  the 

three  numerators  =^x-\-y  -\-  z,  and  the  result  =^  1. 

o     o-      yn    cib I bc^ .  ca 

impiy  (^_^)(^__^)-l-(^__^)(^_^)"^-(^__^)(^_^y 

The  L.O.M.  of  denominators  is  evidently 

(a  —  h)(b  —  c)  (c  —  a). 

This  gives  for  numerator  of  first  fraction  —  ah{a  ~-  h) ; 
and,  by  symmetry,  the  other  numerators  are 
—  hc{h-—c),  —  ca{c  —  a). 

Hence,  we  have  -  ah{a-h)  +  bc{b  -  c)  + ca{c- a) 
{a  —  o){o~  c) (c  —  a) 

_      (a  —  h)(h  —  c)  (a  —  c)  __  -t 
(a  —  b)(b  —  c)  (c  —  a) 

3.    Keduce  the  following  to  a  single  fraction : 


{a  —  h)(a  —  c)  {x  —  a)      {b  —  a)  (b  —  c){x  —  b) 

+ 2 -. 

(c  —  a){c  —  b)  (x  —  c) 


152  FRACTIONS. 


Here  the  L.C.M.  i^{a—h){h—c){c—a){x-a)(x~h){x—c)\ 
the  numerator  of  the  first  fraction  is 

—  a(h  —  c)(x  —  h)  (x  —  c), 

and,  therefore,  by  symmetry,  that  of  second  is 

—  h(c  —  d)(x  —  c)(x  —  a), 
and  that  of  third  is 

—  c(a  —  h)  (<)c  —  a)(x  —  h); 
and  their  sum  is 

—  [a(h  —  c)  (x  —  b)  (x  —  c)  -\-h(c  —  a)(x-~  c)(x  ~  a) 
-}-  c(a  —  h)(x  —  a)(x  —  b)]. 

This  vanishes  if  a  =  b;  hence,  a  —  b  is  a  factor,  and 
therefore,  by  symmetry,  b  —  c  and  c  ~  a  are  also 
factors.  Now  the  product  of  these  is  of  the  third 
degree,  while  the  whole  expression  rises  only  to  the 
fourth  ;  hence,  x'^  cannot  be  involved.  The  other  fac- 
tor must  therefore  be  of  the  form  mx  +  n,  in  which 
in  is  a  number. 

To  determine  n,  put  x  =  0,  and  the  expression  becomes 
abc (a  —  b-\-b  —  c-\-c~-a)==0]  hence,  ?2  —  0,  or  the 
other  factor  is  mx. 

To  determine  m,  put  a  =  0,  b  =  1,  c  =  —  1,  and  m  will 
be  found  to  be  1.     The  numerator  is,  therefore, 
x(a  —  b)(b  —  c)(c  —  a)y 
and  the  result  is 


(x  —  a)(x  —  b)  {x  —  c) 
Simplify 


a-\-b  . b  +  c ,  c -\- a 


{b  —  c)(c  —  a)      (c  —  a)  (a  —  b)      (a  —  b)(b  —  c) 

L.C.M.  of  denominators  is  (a  —  b)(b  —  c)  {c  —  a)\  hence, 
first  numerator  is  a^  —  b'\  and,  by  symmetry,  second 
numerator  is  b"^  —  &,  and  third  numerator  is  c^  —  c^  \ 
the  sum  of  these  =  0,  which  is  the  required  result. 


FRACTIONS.  153 


4.    Keduce 


2       ,2^2,  {x-yy  +  {y~zY  +  {z-xy 


1.    X 


^-y    y-^    ^-^        {^-y){y--^){.^-^) 

Here  the  numerator  becomes 

2{y-z){z-x)  +  2{x-y){z-x) 

+  2{x-y){y-z)  +  {x-yy+{y~zY+{z--xY, 

which  is  evidently 

[{x-y)  +  {y-z)  +  {z-x)J  =  Q. 

Observe   that   the  denominators  become  the  same  by 
changing  the  sign  between  the  fractions,  and  that 
the  expression  is  symmetrical  with  respect  to  a  and  b. 
The  numerator  of  the  first  fraction  is 
a''  +  6  aW  +  12  a'b'  +  8  a'b', 
and,  by  symmetry,  that  of  the  other  is 
~b'''~6  bV  -  12  b'a'  -  8  ¥a\ 
Their  sum  is,  therefore, 
a}-^  -h''  +  6  aW  (a«  -  ^«)  -  8  aW  (a«  -  b^) 
=  (a'  -  b')  (a'  +  b'  +  6  a'b'  -  8aW) 
=  (a^  -  b')  {a'  -  bj  =  (a'  +  ¥)  (a^  -  bj, 
and  since  the  denominator  of  the  given  expression  is 
(a^  ~  b^y,  therefore  the  result  is  a^  +  5^ 

Ex.  47. 

Simplify  the  following : 

x-\-y  )         \x  +  y  ) 

\  a  —  b  J         \b  —  a  J 

a-\-  b I b  -}-  c I c-{-  a 

(b  ~  c){c~  a)      (c  ~  a)(a—  b)      (a  —  b)(b  —  c) 


154  FRACTIONS. 


•  + 


{a  -  h) (a  -c)      (b  -a)(b-cy  (c-  a) (c  -  b) 

a  —  b  .  b  —  c  .   c  —  a  ,   (a  — b)(b  ~  c)(c  ~  a) 
a-^b      b-^  c      c-\-  a      {gl  -^  b^{b  ^  c){c  -\-  a) 


+  • 


(a-\-b){a-\-c)(^x-^ay  {a-^b){b~c)(x-\-b) 


& 


9. 


(a  -\-  c)(b  —  c)  {x-{-  c) 

(x-y){x-z)      {y-x){y-z)      {z-x){z~y) 

(a  —  b){a  —  c)      (b  —  a)(b  —  c)      (c  —  a)(c  —  b) 
1.1.1 


i-oe-')  (;-')(;'')  e-')(i-' 


10.    x^ 


'x'~-2y'\\     ,f2x'-f 


11. 


1 


(^)  +  c  -  2  a)  (c  +  a  -  2  Z^)      (c  +  a  -  2  ^)  (a ,+  ^>  -  2  ^) 
+ 1 


13. 


(h  +  cy'^(c  +  ay^(a  +  by 

a^ ^ P 

(a  —  b){a  —  c)  {x  —  a)      (b  —  d)(b  —  c)  {x  —  b) 

4 


{c  —  a){c  —  b)  (x  —  c) 


(:r-3/)(2;-:r)      {y-z){x-y)      {z-x){y-z) 


RATIOS.  155 


{a+hY  +  (b-cf  +  {a  +  cY 2_. 2_         2 

(a  +  ^)  (^  —  <?)  (<3^  +  ^)  a  +  ^      h  —  c      a-{-b 


15. 


le.     .       \      ,.  +  ^.-4. .+  ■         ^ 


x(x  —  a){x—b)      a(b  —  a)  (x  —  a)      b(b  —  a)  (x  —  b) 


Ratios. 

§  37.  If  ?  --,  therefore,  ad=bc. 
b      a 

Now,  dividing  ad=  be  by  ca,  we  have  -=  -  (1) 

ad  —  be  by  cc?,  we  have  -^=  -  (2) 

c      d 

ad  —  ^6?  by  aZ),  we  have  -  =  ~  (3) 

Also,  — ^-i — -  =  each  of  the  given  fractions.  (4) 

nib  +  nd 

77lb  [  7  )  +  ^^C?  [  -  )         ( 7r<i>  +  7ZC? )  - 

-p      ma  +  y^g  _       W W ^ 

m6  +  nd  "i^^b  +  n(i  7?i6  +  nd 

a         c 

A  very  important  case  of  this  is  7n  =  l,  n  =  ±:l;  hence, 
a c  __a-\-  c a  —  c 


b      d     b+d     b~d 


(5) 


Also^^^  (6) 

a+b      e+d  ^ ^ 

For,  by  (2)  and  (5), 

a  _b  _a~b  _a-\-b         .  a  —  bc  —  d 
e       d      c  —  d      c  +  d'       '  '  a  +  b      c  +  d 

Or  thus  : 

a  —  b__b  d  e  —  d 


a  +  b      a_j_^      c_j^^      c  +  d 
b  d 


156  RATIOS. 


Generally,  to  prove  that,  if  7  =  -,  any  fraction  whose 

0      a 

numerator  and  denominator  are  homogeneous  functions  of 
a  and  h,  and  are  of  the  same  degree,  will  be  equal  to  a 
similar  fraction  formed  with  c  instead  of  a,  and  d  instead 
oih. 

Express  the  first  fraction  in  terms  of  -,  and  for  -  substi- 
c  ho 

tute  its  equivalent  -,  and  reduce  the  result. 

By  (2),  the  fractions  may  be  formed  of  a  and  c^  and  h 

and  d. 

jr-a      c       e    ma -\- nc -\- pe      ace  r>^\ 

11-  =  -==-,  !- ^^^-— =  -  or  -  or  —  (7) 

b      d     f    mh  +  nd+pf      h      d     f  ^  ^ 


mbl  -)  +  nd 


mu  -|-  nc  +  2^^  _       W v^ 

mb  +  nd  -f-  pf  "rnb  +  nd  +pf 

{mh^nd-^pf)- 

b      a 


^^(7) 


nih^nd-\-pf      h 

Tc  a      c        ^  m      p 

If  -—  -  and  —  — -C-, 

b      d  n       q 

ma  z^pc pa  rb  mg mxi       pa  ^r>\ 

nb  zh  qd       qb  ±1  nd       nb  qb 

T^      ma pc^ ma_±^  1      /p-x 

'  nb       qd     nb  ±qd 

pa m£ pa  rb  mc 

qb      nd       qb  ±  nd 

But  — r==-^,  hence  the  equality  stated  in  (8). 
nb     .  qb  ^        J  w 

h      d     f  n      q      s 

ma  :^pc  ±1  re  _pa  zhrczt  me  _        _  ^^  _        /q\ 
nb  ±2  qddzsf       qb  zt  sdzhnf  nb 


RATIOS.  157 


If  an  upper  sign  be  taken  in  a  numerator,  the  corre- 
sponding upper  sign  must  be  taken  in  tlie  denominator ;  if 
a  lower  sign,  the  corresponding  lower  sign ;  otherwise,  all 
the  signs  are  independent  of  each  other. 

Examples. 

1 .    if  -  ^  -,  show  that —-  ^ -— • 

b      d  1a  +  5h      7c  +  5d 


The  given  fraction  = 


5^-4      5^-4 

_^ ^_d ^5c-4:d 

7^  +  5      7^  +  5      7c.  +  5cZ' 
b  d 

2.    If  -  ^  -,  show  that  ^-,^--^3  ^  3-,^-^, 

Dividing  the  given  fraction  by  b^,  we  have 

4+4 


3^-4 


and  this  becomes,  on  substituting  for  -  its  equal  — , 

b  d 

^3  2 

c}  ± I-  3  — 

^d''^    d'_2c'+Sc'd 
3_^_4        Sc'd-4:d'' 
d' 

3.    If  3a  =  25,  find  the  value  of    ^'  +^' 


a'b  -  ab' 

— -  +  1  j  ^  [  —  —  -  J  [by  dividing  both  numera- 
tor and  denominator  by  h^\  But,  from  the  given 
relation 

-  =  -,  we  have,  by  substituting  for  ■-, 
(^  +  l)-^(|-|)  =  35^(-6)  =  -5f 


158  RATIOS. 


li  -  =  -,  prove  that    ,    '    ,,  X  -=    — ^  )  • 
We  nave  -  —-j=  — —.' 


and  this  multiplied  by  -gives  —  =  (      '      \  • 
^  ^  d^         d"      \c  +  dj 

x^  —  ax^  -{-bx-\-ccr  —  ax^b  a 

Multiplying  both  terms  of  second  fraction  by  x,  it  be- 
a^  -f~  aa^  —  hx  . 
x^  —  ax'^  +  bx 
now  each  of  the  given  fractions 

difference  of  numerators    c -, 

difference  of  denominators      c 
Hence,  x"^  -{-  ax  —  b  =^  x"^  —  ax -\- b  or  2ax  =  2b. 
b 


Therefore,  x  =  - 


a 


Iff^g^g  show  that  f  +  "^  +  ^^^^'  +  "'  +  < 
b      d    f  bd+df+fb      b'  +  d'+f 

y      ac ce  __  eg  _  ac  -\-  ce -{-  ea 

^^  bd  ~  'df~  fb  ~  bd+  df+fb 
By  (7)  making  m=^n  =^p  =  1. 

Also^'  =  ^-^  =  ^-+^^i^.  By  (7). 

But  7-:==—,  hence  the  required  equality. 
bd     b^ 

The  problem  is  a  particular  case  of  (9),  with  all  the 
signs  +,  and  a  for  m,  b  for  n,  c  for p,  etc.     . 


RATIOS.  169 


If  the  fractions  given  equal  to  one  another  have  not  monomial 
terms,  instead  of  seeking  to  express  the  proposed  quantity  in  terms 
of  one  fraction,  and  then  substituting  an  equivalent  fraction,  it  is 
often  better  to  assume  a  single  letter  to  represent  the  common  value 
of  the  fractions  given  equal,  and  to  work  in  terms  of  this  assumed 
letter. 

7    Tf     ^"1~  ^     —     ^  +  ^     _    c-\-a 
'       ?>{a-b)~  ^{h  -  c)~  b{c  -  a) 

prove  that  32  a  +  35  5  +  27  c  ==  0. 

Assume  eacli  of  the  given  fractions  =  x,  so  that 

a-\-h  =  ^{a  —  h)x,  h-\-c=^^(b  —  c)x,  c-\-a='b{c  —  a)x, 

a-\-h  .  h  -\-  c  ,  c-\-a        ,       7,7         ,  x     c\ 

or  — ^ 1 ^ 1 ~=^x{a—b-\-o  —  c-\rC—a)^\). 

O  4b  O 

Hence,  adding  these  fractions,  we  have 
32a  +  35^  +  27^  =  0. 

This  example  might  also  be  worked  as  a  particular  case 
of  (7);  thus, 

aAr'b     _    h-\-c    _    c  +  a 
Z{a-b)      4:{b-~c)~b{c-a) 
_2Q{a+h)  +  lb{h  +  c)  +  l2{c+a)  _?>2a  +  ?>bh+21c 

m{a-~h)+m(b-c)+mlc-a)  o 

Hence,  32a  + 35^>  +  27g  ^  0  X     ^  +  ^    =0. 

?>{ar-h) 

Transposing  terms,  etc.,  we  have 
h"       bd       d'-'^  f       df'^  d}~    ' 

"(f-|)"+(7-|)'-»^ 


160  RATIOS. 


That  is,  the  sum  of  two  essentially  positive  quantities 
=  0  ;  therefore  each,  of  them  must  =  0.  Hence, 
we  have 

a      c       r.        J  e      c       r, 
---=0,and^--  =  0; 

hence,  f  =  ^=^  therefore,  g-' =  ^' +  ^' +  < 

Also  f  =  P^-,  tence,  g  =  (p^^)\  ; 
b      b  +  d+f  'b-'      \b  +  d+f) 

therefore,  ('^+^Y=  f  +  f  +  f ■ 


Ex.  48. 

1.    11 -  =  —,  prove  that  —  ' 


b      d'"  ab-ib'  cd~4:d' 


2.    If-  =  —,  prove  that 


prove 


/a  +  cV 


6      cZ'^  W-d'^      \b-d) 

3.  Given  the  same,  show  that  each  of  these  fractions 

\\b-'+d-'J 

4.  If  2:27  =  3  y,  write  down  the  value  of 

2x'~x'y  +  f   ^^^  ^^  x'-Sx^y  +  2y\ 
x^y  +  xy^  +  2  y^  (a;''^  —  y^)^ 

5.  If- =  -=:-,  show  that- =  -— ,     ^ / 

b      a      f  0      mo  —  nd  —pj 

6.  From  the  same  relations  prove  that 

—  —  /o^  —  ^g— '^^Y 
h^      \h  —  md—nfj 

7.  Ifl±^^^fl+£i±4\then^  =  (&-a)H-(5  +  a). 


RATIOS.  161 


8.  If  -^^^^ — ^ — /  ^  \y 1  =  a,  prove  that  x  — 

•\/(a  + x)—-^{a-—x)  1  +  0^ 

^     jn  mx  -j-  a-j-  b      mx  —  c  —  d  ,i,  b  —  c 

9.  li ■ ■ —  = -,  prove  that  x  = 


10.    If 


nx-{-a-\-  c       nx  —  b  —  d'  n  —  m 

a~b    b  —  c   c  —  a   __     a-\-  b  -}-  c 


ay  -{-bx      bz-\-  ex      cy  -\-  az      ax  -\-  by  -{-  cz 

then  each  of  these  fractions  = , 

a~\-b  -\-  c  not  being  zero. 

a  —  b      2{b  —  c)      6(c  —  a) 

12.  If  V«+V(«-^)  _  1_  3i,ow  that  ^^^  =  (^^\ 

-\Ja  —\/{a  —  X)      a  a         \l-{-  aj 

rp^ oi'y  01    'Y'^ 

13.  If  —- — ^—-  =  -^ -,  and  x,  y,  z  be  unequal,  show 

that  each  of  these  fractions  is  equal  to  x  -{-  y  -{-  z. 

14.  If  ^'  +  2^  +  1  ^  y^  +  2y  +  1  g^Q^  ^^^^  ^^^^  ^f  ^^^g^ 

:r^  — 2;r  +  3      y^  — 2y  +  3 
fractions  =  {xy  —  1)  -=-  (xy  —  3). 

15.  If25^--16^y-4),3i,,^that^+l=§. 

10a; +  8  2rc-4  a; +5      5 

•^      6  +  c  y-2b     y~2o 

17.  If  Y^!+i.VV^^±^1  =  Y^^^^\  prove  that 

25a^  +  276'  +  22c^  =  0. 

18.  Tf      a'     _      ^'     _      c'     ^ 


o;^  —  yz      y  —  2;;r      z""  —  a;y 

show  that  a^x  +  Z>^y  +  <?'^2;  =  (a^  +  S'^  +  c^)  (a;  +  y  +  2). 


162  RATIOS. 


19.     If  ^^ =7—^ = ^ r 

a-\-h  —  c      0  -\-  c  —  a      c  -\-  a  —  b 

then  will  {a  —  h) x  -\-  {h  —  c)y  -\-  {c  —  a) z  =^  0. 

20     If  ^ -  ^  -  1    then  (ci'  +  c'  +  e'y_  a' +  c' +  e' 

o-i      Tf  i^-^  +  <^y  _  gy  +  ^:3  _  az-{-  ex 
a  —  h  b  —  c  c  —  a 

show  that  {a -\- b  -\-  c)  {x -{- y  -\- z)  ^=  ax-\-by  -\-  cz. 

22.  If  ^'  -  ^^y  -  ^"  +  ^^f  :==  ^ZL^^  siiow   that   each   of 

^•^  +  x^a  +  :ra^  +  <^         x-\-a 
these  expressions  =  1. 

23.  ul(^)  =  l(^-^)^^('-^,  and  «,  ^,  .  be 

6\a  +  bJ      b\b  +  cj      10\c  +  a) 

unequal,  show  that  16a+ll^  +  15c^=0. 

24.  If  (^^±y^=ljzy^  prove  that  x'+y''+z^  +  2xyz^l. 

\y  +  zxj      1  —  x^ 

25.  If —^==_^z::r_^,  show  that  a  +  5  +  c=:0. 

x  —  y      y  —  z      z  —  X 

26.  If  f  =  4,  prove  that  ^  ==  VM +V(^^). 

b      d^  a-b      ■yj(ac)—yj{bd) 

27.  If  ?  —  -,  =  ^,  then  each  is  equivalent  to  ^  ~^  "^Oj]-  ne 

b      d    f  ^  Ib  +  md  +  nf 

Hence,  show  that 

a b c 

2z  +  2x-y~~2x  +  2y  —  z~2y  +  2z  —  x 

when —, ==  — - — ^ 


2a  +  2b-c      2b  +  2c-a      2c  +  2a-b 

28.    If  f  =  £   prove  that  f ^V^  J^^^\ 
h      d^  \c-dj      Wc^^'  +  d^^) 


RATIOS. 


163 


y^ 


I 


prove  that  -  (y  —  2j)  +  '^  (2;  —  ^)  +  -  {x  ~y)  =  0. 


Ix (ny  ~  mz)      7)iy  {lz~  nx)      nz  {mx  ■—  ly) 
my 


i]iQnWi\\^(l-x)  +  ^(m~y)  +  -(n~z)^0. 
Ix^         ^      my^         -^^     nz^         ^ 


31.    it  2;  = :: ,  and  y  = » 


y 

show  that  X 


■yj{az^  —  a^) 


32. 


rr  x^  —  yz  _y'^  —  xz  ^z"^  —  xy _  ^ 
7^~  /Ta      ~      7^      "  •*■ ' 


show  that  x-{~y  +  z  = 


a^x  +  hhj  +  ch 


a'+b'  +  c'' 


33. 


jnm      n      r       j  ^       y       2;'      ., 
It  —  =  -  =  -  and  -5  =  7i  =  -2  —  1, 
X      y      z  a^      Ir      c^ 

prove  that  _  +  -  +  -  =  3  ^,^^,^^, 


n,3n .  ^3n 


34. 


^3n  _  ^n^ngn_(-^n_^n_^gn^^3 


OK        Tf  ^1 ^2 .^3 ^'n 

^1  ^2  ^3  ^n 

bA-hA+ (-ly-'b^-A 


^1  V^2^3  +  ^2  V6364  +  • 


164  COMPLETE    SQUARES. 

36.    li4+A±G^A  +  ^  +  '^, 
abc  a       b       c 

^nd.  {A  +  B  +  C){a  +  h  +  c)  =  Aa+  Bh  +  Cc, 

then  will  -^  +  -^,  +  -^  =  0, 
1  +  a^      1  +  0^      1  +  r 

and  also -| -\ =  =  0. 

a  +  i      5  +  i      c+^- 
abc 


Complete  Squares,  Etc. 

What  quantity  must  be  added  to  x^  +px  to  make  it  a 
complete  square  ? 

Let  r  be  the  quantity. 

Then  r^  +^;r  +  r  =  complete  square  =^{x+  -y/ry 
=^x'^  +  2x-y/  r  +  r. 

Equating  coefficients,  we  have 

2^T  =  p\  hence,  r  ="^  =  (-^  I  • 
V        ^ '  4      \2J 

Or  thus :  Since  (a  +  ^Y  =^a^-\-2ax-\- x^,  we  observe 
(see  §  12)  that /owr  times  the  'product  of  the  extremes 
is  equal  to  the  square  of  the  mean  ;  hence, 

4a;V  =  pV  ;  therefore,  r  =  [^  )  ,  as  before. 


COMPLETE    SQUARES.  165 

Or,  we  may  extract  the  square  root  and  equate  the 
remainder  to  zero  ;  thus  : 

P 


-\-px  -{-r  {x  +  ^ 


2x-{-^       px-{-r 


P 

Now,  if  the  expression  be  a  complete  square,  this  re- 
mainder must  vanish ;  hence,  we  have 

2.  Find  the  relation  connecting  a,  h,  c,  if  ax^  -{-hx^  c  is  a 

complete  square. 
Assume  ax"^  -\-hx-{-  c  =  (-y/ci  •  x  +V^)^ 

=  ax"^  +  2->/(a<?)  X  +  c. 
Now,  since  this  holds  for  all  values  of  x,  we  have 

2-^(ac)  =  h,  or  b"^  =  Aac,  the  relation  required. 

3.  Determine  the  relation  amongst  a,  b,  c,  in  order  that 

a^x"^  -}-  bx  ~\-  be  -{-  b"^  may  be  a  perfect  square. 
As  in  Exam.  1,  we  have  AaV  (be  -\-  b"^)  =^  bV  ; 

hence,  -—--=:  I. 
4a^      b 

Or  thus : 


Assume  aV  -\-  bx  -{-  be  -{-  b^  =  (ax  +  V^c  +  b'^y 

=  aV  +  2  ax\^be  +  b'  +  be  +  b\ 
Equating  coefficients,  we  have  b  '=2a^be  +  b"^ ; 

hence,  — —  —  ~=zz\    as  before. 
4a^      b 


166  COMPLETE    SQUARES. 

The  same  result  may  also  be  obtained  by  extracting  the 
square  root  and  equating  the  remainder  to  zero. 

4.    Show   that  if  x^  +  ax^  ■\- hx^  -\-  ex  •\-  d  be   a   complete 
square,  the  coefficients  satisfy  the  equation 
c^  -  aH  =  0. 

Is  it  necessary  that  the  coefficients  satisfy  any  other 
equation  ? 

Extracting  the  square  root  of  x^  +  ax^  +  hx"^  -\-  ex  -\-  d  in 
the  usual  manner,  we  have  for  the  final  remainder 


['-K-Dl'+^-IMl* 


Now,  if  the  expression  be  a  complete  square,  this  re- 
mainder must  vanish  ;  and,  that  it  may  vanish  for 
general  values  of  x,  we  must  have 

.-|(*-f)  =  0  (1) 

<'-l(»-i7"°  ^^' 

Eliminating  ^  —  -r ,  we  have  c^  —  d^d  =  0  (3) 

The  coefficients  must  satisfy  the  equations  (1)  and  (2), 
and  therefore  either  of  these  equations,  together  with 
the  equation  (3),  which  results  from  them. 

The  same  result  may  be  obtained  by  assuming 
x'  +  aTf  +  hx^  +  cx+d=  {x^  +  \ax  +V^)' 
=  x'  +  ac(^  +  2x^^d+\o?x'  +  ax-^/d+d. 

Equating  coefficients,  we  have  2^d-{-\a'^  =  h  (1) 

and  a^d  =  c  (2) 

From  (2)  we  have  c^  —  a^d  =  0,  as  before. 


COMPLETE    SQUARES.  167 

5.  What  must  be  the  value  of  m  and  of  n 

if  4  a;*  —  12  :r^  +  25  r?;^  —  4  mx + 8  n  is  a  perfect  square  ? 
Assume  the  expression  ^[i^^x^  —  ^x  -\-^{^n)f 

=  4:x'  -  12x'  +  4:x'  ^(Sn)  +  dx'  ~  Qx-y/{^n)  +  8n. 
Equating  coefficients,  we  have  6-^/(892)  =  4m  (1) 

and  4V(87z)  +  9=-25  (2) 

Therefore,  n  ^=  2,  7?i  =  6. 
Or  thus  :    Extracting  the  square  root  in  the  ordinary 

way,  the  remainder  is  found  to  be 

(—  4m  +  24) :i'  +  8n  —  16  ;  hence,  we  must  have 

47?2  +  24  —  0,  or  m  =  6  ;  and  8n  —  16  —  0,  or  w  =  2. 

6.  If  ax^ -\- hx"^ -\~  ex -\- d  be  a  complete  cube,   show  that 

ac^  =  dh^  and  ^^  ==:  3  ac. 
Assume  ax^  -{- bx"^  +  ex -j~  d  =  (a^x  +  d^y 

=  ax'  +  S  aW^x"  +  3  md^^x  +  d. 
Equating  coefficients,  h  =  ?>aM^  (1) 

c^'^aM^^  (2) 

Dividing  (1)  by  (2),     -==  -77  ;  hence,  ac^  =  db^. 
c      as 

Also,  b'  =  9aMi  (3) 

Dividing  (3)  by  (2),   -  =  3a  ;  hence,  b'  =  Sac. 
c 

7.  Find  the  relations  subsisting  between  a,  b,  c,  d,  e,  when 

ax*'  +  bx^  +  cx'^  -j-  dx -\~  e  is  3i>  complete  fourth  power. 
Assume  ax^  +  bx^  -{-  cx"^  ~\-  dx -\-  e  =  (a^x  +  e^y 

=  ax^  +  4  ah^x^  +  6  cae^x"^  +  4  a^e*.?;  +  e. 
Equating  coefficients,  we  have 
^  —  4  ai^i, 
c  =  6  aM, 
c?=4a^ei; 
whence,  5c?— 16ae  (1) 

bc:=  24:a^ei  =  6a  X  4aiei=::  6 ad      (2) 
cc?--24al^--6c  X4aic4^65e       (3) 


168  COMPLETE    SQUARES. 

8.  Show  that  x^  -{-px^  +  qx"^  -{-rx  -\-  s  can  be  resolved  into 
two  rational  quadratic  factors  if  s  be  a  perfect  square, 
negative,  and  equal  to 

p^  —  ^q 
Since  —  5  is  a  perfect  square,  let  it  be  ?^^. 
Assume  ^*  -{-px^  +  <J[^  -\-rx  —  n^ 
=  (x^  +  mx  +  n)  {x^  +  m^x  —  n) 
=  :?;*  +  {m  +  vi^)  x^  +  mw}x^  —  n  {m  —  ?7i')  x-n^. 
Equating  coefficients,  we  have 

7n-\-7n'  =Pj 
mm'  =  q, 

n 
m^  +  2  mm'  +  m'^  ='p'^, 
4:mm'  =  4:q. 

Hence,  (m  —  m')^  ==  p^  —  4  <7  —  — • 

n 


Therefore, 


r 


p^  -  Ap' 


Ex.  49. 


1.  What  is  the  condition  that  (a  —  x)(h  —  x)  —  &  may  be 

a  perfect  square  ? 

2.  Find  the  value  of  n  which  will  make  2^^  +  8:r  +  ri  a 

perfect  square. 

3.  Find  a  value  of  rr  which  will  make 

x^  -\-^x^  -\-\\x^  '\-?)X-\-^\  a  perfect  square. 

4.  Extract  the  square  root  of 

{a  -hy-2  {o?  +  h-")  (a  -hy  +  2  {a'  +  h'). 

5.  Find  the  values  of  m  and  ?2  which  will  make 

4:X*  —  4:X^-{-5x^  —  7nx  +  92  a  perfect  square. 


COMPLETE    SQUARES.  169 

6.  What  must  be  added  to  x'—.^(4:x'-16x'+lQ)-4cx^ 

in  order  to  make  it  a  complete  square  ? 

7.  The  expression  x^  -j-  x^  —  16  x"^  —  4:X  -}-  4:8  is  resolvable 

into   two   factors   of    the   form   x"^  +  mx  +  6    and 
x"^  -{-nx-\~  8.     Determine  the  factors. 

8.  Find  the  value  of  c  which  will  make 

4:X^  —  cx^  -\-  5  x"^  -{-  —  -{- 1  a  complete  square. 

A 

9.  Obtain  the  square  root  of 

10.  If  (a  —  h)x^-\-{a-^  hjx  +  (a'  -  y)  (a  +  Z>)  be  a  com- 

plete square,  then  a  =  3^,  or  Z>  =  3a. 

11.  Find   the  simplest  quantity  which,   subtracted   from 

aV +4  a5^+ 4  acx  +  5  5c  +  b'^c'^,  will  give  for  remain- 
der an  exact  square. 

12.  x^  —  4: x^ —- x"^ -\- 16 X  —  12  is  resolvable  into  quadratic 

factors  of  the  form  x"^  -f  mx  -{-p  and  x"^  -{- nx -]-  q  ; 
find  them. 

13.  Find  the  values  of  m  which  will  make  x'^  -f  max  +  a^ 

a  factor  of  x*  —  ax^  -f  a^x"^  —  a^x  +  a*. 

14.  Show  that  if  ^*+  ax^-\-hx'^-\~cx-\-d  be  a  perfect  square, 

the  coefficients  satisfy  the  relations  8c  =  a(4:b  —  a^) 
Sind64:d=(4b-a'y. 

15.  Investigate  the  relations  between  the   coefficients  in 

order  that  ax"^  -f  5y^  +  cz^  +  ^^y  +  ^3/^;  -\-fxz  may  be 
a  complete  square. 

16.  If  r^^  +  ax"^  -i-bx-\-  che  exactly  divisible  by  (x  +  dy, 

show  that  i (b' -d'')=-  =  d{a-2 d). 

17.  Determine  the  relations  among  a,  b,  c,  d,  when  * 

ax^  —  bx^  -\-  ex  —  d\^  2.  complete  cube. 


170  COMPLETE    SQUARES. 

18.  The  polynome  ax^  -{'?>bx^ +  ?>cx  +  d  is  exactly  divis- 

ible by  {a  —  xy  ;  show  that 
(ad  -  bey  =  4  (ac  -  h')  (hd  -  c"), 

19.  Find  the  relation  between^  and  q  when  x^ -{-px'^ -\-  q 

is  exactly  divisible  by  (x  —  ay. 

20.  If  x"^  +  nax  +  a^  be  a  factor  of  x^-\-  ax^-\-aV-\-  a^x  -\-  a*, 

show  that  71^  —  n  —  1  —  0. 

21.  If  x^  +  ax^  +  hx'^  -{-  ex  '\-  dhoi  the  product  of  two  com- 

plete squares,  show  that 

(4Z)-a7  =  6H  {^h'-d')a  =  ^c,  a-yy(Sa'-2h)  =  Sb. 

22.  Prove  that  ri;"^ +  »^^  +  5':^:''^  +  ro; -f  s  is  a  perfect  square 

if  p'^s  =  r^  and  q  =^+  2y^s. 

23.  If  aa^ -}-Shx'^-{-Scx-\-d  contain  ax"^ -{-2hx-{-c  as  a  fac- 

tor, the  former  will  be  a  complete  cube,  and  the 
latter  a  complete  square. 

24.  If  mV  -f  ^rx;  -\- pq  -{-  q"^  he  a  perfect  square,  find  p  in 

terms  of  m  and  q. 

25.  Find  the  relation  between  p  and  q  in  order  that 

x^  -\-px'^  -{-  qx-{-r  may  contain  {x  +  2)^  as  a  factor. 

26.  li  x^  -\-px^  -{-  qx-\-r  be  algebraically  divisible  by 

3:^;^  +  2px  +  q,  show  that  the  quotient  is  ^  +^- 

o 


CHAPTER  V. 

Linear  Equations  of  One  Unknown  Quantity. 

§  38.  Preliminary  Equations.  Although  the  follow- 
ing exercise  belongs  in  theory  to  this  chapter,  in  practice 
the  numerical  examples  should  immediately  follow  Exer- 
cise I.,  and  the  literal  examples  Exercise  HI.  Like  those 
exercises,  this  one  is  merely  a  specimen  of  what  the  teacher 
should  give  till  his  pupils  have  thoroughly  mastered  this 
preliminary  work.  But  few  numerical  examples  are  given, 
it  being  left  to  the  teacher  to  supply  these. 

Ex.  60. 

What  values  must  x  have  that  the  following  equations 
may  be  true  ? 

1.  x-b-=0',  x-?>\  =  0]  x~a  =  0]  x  +  ?>^0. 

2.  x  +  ^  =  0-^  x  +  a  =  0\  a;  +  3==6;  :r  — 4  =  6. 

3.  x~a=^h]  x-\~a^=^c)  x~h  =  ~c]  6  — a:  =  3. 

4.  8-37...  10;  5  +  a;--ll;  9  +  a;  =  4;   1-x^-b. 

5.  8  +  :?;-=  — 6;  a  —  x=^'^h',  2a^x  +  2>h]  ^a  =  bh—x. 

6.  2x  —  ^-=^]  ?>x  +  d>-=20]  ax  =  a^;  mx^hm. 

7.  Sx  =  c;  ax  =  6;  ax=^0]  (a -{- b)  x  =  b -{- a. 

8.  (a~b)x  =  b-a]  (a+b)x=(a+by]  (a~b)x=a^-b\ 

9.  {a  +  b)x=.b'-a']  (a'~ab  +  b')x  =  a'  +  b\ 

10.    (a'~b')x^a~b',  (a' ~  b')x  =^.  a  +  b  ;  (a'  +  b')x=l. 


172  LINEAR    EQUATIONS. 

11.  a-\-x —  b  =  a-{-b]  x  —  a-{-b  =  b  ~ x-\-a. 

12.  2a  —  x  =  x  —  2b]  ax-{-bx  =  c;  ax  —  b^=cx. 

13.  ax~b  =  bx  —  c]  ax  —  ab  =  ac. 

14.  ax  —  a^  =  bx  ~  b"^  ]  ax  —  a^  =  bx  —  b^. 

15.  ax~a^  =  P--bx]    ax -{- b -{- c  =  a -{- bx -}- ex. 

16.  a~bx  —  c  =  b-~ax-\- cx]    a-\-bx-\-cx^-=ax—b-^cx^. 


17. 

bx  ~  cx^  -\-  e  =^  ex  —  b  - 

-ex^]    3X 

18. 

6                     c 

ax^-—' 

0 

19.  a5:r=f  +  ^    lcx^^  +  ^. 

b      a  be 

20.  \x  =  b]   1^  =  8;   0.5^  =  2;   0.3x  =  0m. 

21.  0.02rr-:20;    0.3:r==0.2;   0.4:?;=  0.6. 

22.  0.i8a;  =  1.8;   --5;    ^  -  c. 

a  b 

nc.     ci^ h  ^     _  <^^    _  z. 

6       e      a+b  a+b 


25. 


26. 


29. 


&  — a         a  —  b' 

a+ J        5+a 

a-\-b     a  —  e  ^ 

a-\-e'^      a  +  b' 

1_1.  2_3 

a;      2'   a;      5' 

1_1_.    l_a. 
^      ab'    X      b' 

a      b,    7_1      1_ 
x~  c'   X      3      4 

3    ,    4  _33 

20  "^50;      5x 

5;    -  +  -=0. 
3      x^  c 

5     -f^         7 

■         5-7, 

9 

27.  -=4;   --=T 


28. 


-7      3a; -4  4-3:r 


LINEAR   EQUATIONS.  173 

30.  (x-4:)-{x  +  b)  +  x=^3;  2x-(x-b) ~(4:-Sx)==5. 

31.  2(S-x)+S(x-S)=0',  2(3a;-4)-3(3-4rr)+9(2-^)=10. 

32.  a(l~2x)—(2x  —  a)^l',  x  —  5(a~ x)  ==  bx  —  5a. 

33.  mx(3a~4:)  +  Smx  —  Sa+l  =  0. 

34.  a(bx  —  c)-{-b(cx  —  a)-\-c(ax  —  b)  =  0. 

35.  a  (ax  ~b)-\-b  (ex  —  c)  -\-  c  (ex  —  a)  —  0. 

36.  a (bx  —  a)  -\-  b  (ex  —  b)-\-e (ax  —  c)  —  0. 

37.  a(x-2b)  +  b(x~2c)  +  c(x-2a)==^a''  +  b'  +  c', 

38.  3J3[3(3:r-2)-2]-2|-2=:l. 

39.  9J7[5(3:r-2)-4]-6|-8==l. 

40.  i^i[iab  +  2]  +  2)  +  2]  +  2|  =  l, 

41.  4?|[iab  +  2]  +  4)  +  6]  +  8|  =  l. 

42.  iii[ia^-i)-i]-ii-i=o. 

43.  m[m^-H)-H]-m-n=o, 

44.  llk\[^(f[|^  +  4]  +  8)  +  12]  +  20|  +  32-68. 

45.  tJf[ia[^^  +  7]-3)  +  6]-l}  =  4. 

46.  r\q[p(n[mx-~a]  —  b)  —  c]  —  dl~e  =  0. 

47.  (l  +  6:r)^  +  (2  +  8a;)^==(l  +  10;r)l 

48.  9(2ri:  -  7)^  +  (4a;  -  27)^  =:  13(4:^;+  15)(a:  +  6). 

49.  (3-4rr)^  +  (4-4^7-:2(5  +  4^)^ 

50.  (9  -  Ax)(9  -  5x)  +  4(5  -^)(5  -  4^)  ==  36(2  - xf. 

§  39.  In  solving  fractional  equations,  the  principles  illus- 
trated in  the  sections  on  fractions  may  frequently  be  applied 
with  advantage,  as  in  the  following  cases. 

When  an  equation  involves  several  fractions,  we  may 
take  two  or  more  of  them  together. 


174  LINEAR   EQUATIONS. 


Examples. 

1.    Solve  ^  +  g-^=-i-. 

Here,  instead  of  multiplying  througli  by  the  L.C.M.  of 
the  denominators,  we  combine  the  first  fraction  with 
the  last,  getting  at  once 

7:^-3       7       1 


6ri;  +  2      14      2 


7x  ~3~3x-{-l,  and  x=l. 


2     Solve  ^^  +  ^-  1^^-^  +g^!g-£zLl^,. 
9  17a; -32  ^3      12         36 

In  this  case,  taking  together  all  the  fractions  having 
only  numerical  denominators,  we  get 

8^  +  34+12^-21^  +  rg+16^  13.r-2  . 
36  17^-32' 

0T  —  =  '^^^-^,     .-.  425  r^;  -  800  =  234^^^  -  36  ; 
18      17^-32 

hence,  ^  =  4. 

It  is  often  advantageous  to  complete  the  divisions  repre- 
sented by  the  fractions. 

3.    Solve  4^-17__3|_-22a; 


9  33 


X\  54:  J 


Here,  completing  the  divisions,  we  have 
4^_17_1  ,  2^^     _6     X 
9        9       9"^  3  "    ''^     :r"^9' 

9  9     X 

n 

Therefore,  —  2  =  —  -,  or  rr  =  3. 

X 


LINEAR    EQUATIONS. 


175 


ax  -\- h  .  ex  A-  d  , 
1 1 ^ —  =  a-\-  c. 

X  —  TYl         X  —  n 
am~\-h 


.\a  + 


6. 


7. 


cn  +  d 
+  c  +  —^ —  —  a-\-  c. 


X  —  771  X  —  n 

.*.  (ar)x  +  Z^)  (x  —  72)  +  (en  +  d)  (x  —  m)  =  0. 
.'.  (am  +  ^  +  <^^  +  <^) ^'^^  —  (<^  +  c) 7)in  +  Z)7Z  +  dm. 


5.    Similarly  may  be  solved 


ax  +  h      ex  - 


■  d         ex^  +fx  —  g 


-a-\-  c  -{-  e. 


x  ~vi  '    x  —  n    '  {x  -—  Tu)  {x  —  n) 

am,  ~{~  h  ^  en -\-  d      [e  (m  +  n)  +/] x  —  emn  —  g 


X  —  'in    '    x~n    '  (x  —  7Yi) {x  —  n) 

.'.  (am,  +  h)  (x  —  n)  -{-  (en  -}-  d)  (x  —  m) 
+  [e  (?7^  +  n)  +/]^  —  eTnn  —  ^  —  0. 

.•.  [(a  +  c)m  +  ^  +  ((?  +  e)7i  +  d-\-f]x 
=  (a~\-b  -{-  e)  mn  -\-bn-\-  dm  +  g. 


=  0. 


132^7+1      8:y  +  5. 
3:r+l  :^-l 

.-.44- 


43 


+  8  + 


52. 
13 


3:1' +1 
.•.39.T+13-=43r^ 

25 -i 


x-\ 

43,  and  x  - 


=  52.  or 


13 


43 


1      3a;+l 


14. 


M^±M^5- 


23 


x+l     '     Sx  +  2        '    '  a;+l 

Taking  the  last  fraction  with  the  first,  and  multiplying 
the  resulting  equation  by  15,  we  have 

240:r  +  63_^,^  ,  5^-30 


;.80- 


3x  +  2 
97 


:75- 


x  +  l 


-  75  +  5  - 


Sx  +  2 
97     _    35 
3:^7+2     x+l' 


35 

x+l 

X  =  27,  and  x  =  3f . 


176 


LINEAR   EQUATIONS. 


8      ^~<^    ,    X  —h    ,    X  —  C 


b  -{-  c      a-\-  c      b  -\-  c 
.  x  —  a      1    I  x~b 


3. 


l  +  ^JH^-^l 


^0. 


9. 


10. 


b  +  c  a  +  c  b-\-  a 

'  ^~(<^  +  ^  +  g)  .  X  —  (a -}- b -{- c)  .  rg— (a+^  +  g)_/ 
b  -}-  c  a-\rC  b  ■\- a 

which  is  satisfied  by  a;  —  (a  +  ^  +  ^)  =  0 ; 
.*.  a;  =  a  +  Z>  +  ^. 

m      .      n     _m  -{-n 

x  —  a     x  —  b      X  —  c 

7n(x ■—  c)  ,  n(x  —  c)  , 

.*. — ^^ ^ -) — ^^ —^  =  m-\-n. 

x  —  a  x  —  b 

.Ma^^^njb-cl^^     (See  Exam.  4.) 

X  —  a  x  —  b 

:.  \m{a  —  c)  +  n(b  —  c)]x  =  mb (a  —  c)  -{-  na(b  —  c). 


3Vv-5 


77      9\7/-lOj      18 
12  2  5 


.  1 
**3  '  3/-5  '  6  '  3/-7     9     3/-IO      18 

2/-5     y--7     3/- 10 

2/  -  5     3/  -  7 
.-.  ^:i^  + -^=^  =  0. 

2/-5     y-7 
.•.-82/ +  50-0.     .•.y  =  6i. 

Ex.  61. 

10:r  +  17       12a;  +  2  _  5a;  -  4 


18 


13a; -16 


6a;  +  13      9a;  +  15  .  o^2a;+15 
15  5a;-25'^  5 


4. 


13. 


LINEAR    EQUATIONS.  177 

7a;+l  _  3  5  fx  +  A\   ,  o  1 

4:X-7  ,  2-14:X     d^  +  x ^  10 -3f^ __  19 
2:z:-9  7  14  2  21* 

2x-{- a    ,     Sx  —  a  __  ^^^ 
3(^^^2(^Ta)"    ^' 

^-4    ,  3^-13      1 


6^  +  5      18:2;-6      3 

3:g+  1        ^^-- 11  _  -1 
2:?:-15      2:?;-10~ 

8.    ^-^t^  —  5^2. 


12  _^  .r  —  4  ^  2  -f. 


.T-7      ^-12 
,^     3a:-  19  ,  3a;-ll      n 

10.     : :—  =  D. 


X  - 


13         x+7 


2:z:+l"^3(:r-3)~6 
12.      ^  +  1      I     ^^  +  4    _  9 


4(:i,'+2)      5a; +  13      20 

5(2.^  +  3)      5-7.-^^^_3^ 
2a;+l         2a;-5 


14.    -^  +  .   1  4 


a:  —  7      a;  —  9      a;  —  8 

j^g     7  a: +  55      3>r__g      3a;'^  +  8 

2a;  +  5        2  2a:-4' 

16        1^      I      15     _     32 

07-16  ~^a:-18      a;-17* 


178  LINEAR    EQUATIONS. 

l-25a;       S-2^x   ^28-5.^-      10:?;- 11      x 
15  14(^-1)  3  30       "^3* 

18.       1         2  +  2i^^-^r.^_,^ 2_^ 

:r  —  2         6  —  6r^  +  ;r^  :r  —  5 

^^     30  +  6a;  ■  60  +  8^_    48        ^^ 


20. 


:?;+l  ^+3        x+1 

5x^  +  x-S^7x''-3x~9 
5x  —  4:  7.^—10 


Q  -  5/  •    X         o X  ~Y~  J-     I    ^         O 

x~2     x—7      x~l      ^—6 

22     x'-Sx-9      x'-7x-17^2(x'~6x-lb)^ 
X  —  6  x  —  9  x  —  S 


23. 


24. 


25. 


26. 


27. 


4^  +  7  ■  4:r  +  9^4:r  +  6  .  4rg+10 
4a:  +  5      4;r+7      4:r  +  4       4^+8* 

2:?;-3      2^-4      2:i;-7      2a;  — 8 


2:r-4      2x-b      2:?;-8      2:r  — 9 

7a;  +  6  _  2:r  +  4|-  ,  x_^  Ux  _  a;--3 
28         23:?;- 6      4       21  42 

:?;  — 5,:?;  — 11      X'-7  ,    x—9 


X  —  o       X 


-12      :?;-8      :?;-10 


^-125       2  — 6;r^         5:r-^(10-3a;) 
2  13     ~^  39 


10  — 17;r       .      1  +  ^      ,'7^1 
'    27(:?;^-:?;+l)"^2(a;^  +  l)"^54(:?;+l)      9(0:2+1)* 

2x''+x~30  ,  :r2+4:r-4_:?;'^-17  .  2;2;2+7a;-13 
2a;-7  :r~l  a;-4  "^        2a;-3 


30. 


_a  .  x  —  h  __        (g  —  5y       __  2(a  — 3;) 
■b      x  —  a      (x  —  a)(x  —  b)         a-{-  x 


LINEAR    EQUATIONS.  179 


i 


3^     12.r+10a      28x  +  117a_j^Q 


32     ^H^-^      131:^-11  ^13^:^-7  ■    ISjrX-d 


33. 


1  X  16x 


2(x~iy      2(:^-l)     *2(:r^  +  l)      (x~l)(x'  +  l) 
7|-  —  X     X  (^ 


34.  KI-  +  4)-^V^=|^^-lJ- 

^^     3:r  81:?;'- 9  q         3/2:^'-1\      57-3;r. 

35.  — •  — -^=6x ( 1 

2       (3:^;-l)(:r+3)  2^^  +  3  7  2 

36      1  +   2^+1    _   4:^  +  5   ^   x^-~x-\-'l  _  . 
2(:r-l)      2(a;+l)      :r'^-2^  +  l 

7:^;~3Q      5a;-7      2-21:y 
10|  i-;r-3  21 

.^42:y-171_-^Q  ,     2:^  —  9   _i/4_,7^>v 

33^    18:^-22  1+6.,^  101-54^, 

13-2:^'^       ^       8  ^  8 


37. 


39. 


40. 


4-9:r      5-12:r      o  240:^ 


1-3^       7-4:r  7-25:r  +  12:r' 

8:r  +  25      16^+93__18:r  +  86      6.^  +  26 

2.x'  +  5        2.r  +  ll         2:r  +  9         2x-\-n' 


41.  — i 1 L_+      1      ^0. 

x-^a-\~h      X  —  a  +  h      x  +  a—h      x  —  a  —  h 

§  40.    The  results  deduced  in  tlie  sections  on  ratios  may 
often  be  applied  with  advantage. 


180  linear  equations. 

,  7  Examples. 

ax -\-  0 _7n 

ex  -\-  d      n 

.^  {axArl)d-{cx-\-d)h  ^  md-nh  .p        ^^^. 

'  {cx+ d)a  — {ax-\-h)c       na—mc 

7)id  —  nh 

:.x  = ♦ 

na  —  mfic 

^      ax^  -\-hx  -\-  c       a 

/O. =1  • 

rax  -\-  nx  -{-p      m 

{ax^  -{-hx-\-  c)  —  ax^  __  a 
(vix^  -\-nx  -]-  p)  —  rrix^      w. 

bx4-c       a      , 

.*. ' —  =  — ,  etc. 

nx  -\-p      m 

Sx+7      Sx~13 


3. 


4. 


(Page  155) 


07  +  4  X  —  4: 

By  (5),  page  155,  eacli  of  these  fractions 
difference  of  numerators 


difference  of  denominators 

"8        x  +  4c  2:  +  4'^^2     ^  +  4'"^ 

mx  ~\-  a-{-b  _  mx  -{-  a-{-  c 
nx  —  c  —  dnx  —  h  —  d 

mx  +  a  +  h      nx  —  c  —  d         i.     /o\  ^KK 

.  . ;      T    = 7 — :^;  or  by  (2),  page  155, 

mx  +  a  +  c      nx  —  b  —  d 

mx  +  a  +  b  _nx  —  c  —  d ^ 
b  —  c  b  —  c 

01  (n  —  m)x^^a-\-b-{-c-{-d\  .'.x= 

1,2,3  6 


6a     x-{-da     x  —  2a     x- 


LINEAR   EQUATIONS.  181 


Transposing, -— 

x  —  oa 

+ 

2 
:i^  +  3a 

6              3 

x  —  a      x  —  2a 

Sx-9a       _ 

3:^-9 

^                                    ^           \.r.rK\r.4^^Q^ 

"  x'-Sax-lSa' 

x' 

-3a:r  +  2a^     2Qa^^' ^'^^''''^" 

.\Sx~9a  =  0. 

.-.  x--=  3a. 

Ex.  52. 

1. 

1  +  :r  _  1 

1—x      a 

8. 

X  -{-7)1 a  +  b 

x—m      a—b 

2. 

x~a 

9. 

a  +  b        a  —  b 
l-\-  ex      1  —  ex 

3. 

ax-\-b  _m 
ax  —  b      n 

10. 

a  ~\~  bx  _  c -{-  dx 
a  +  b        c  +  d 

4. 

a  +  x       ^ 
b  +  2x 

11. 

a  +  bx      e  +  dx 
a  —  b        e  —  d 

5. 

a{h  +  x)_^ 
a  —  x 

12. 

a—x      a+x 
b  —  X      b  +  X 

6. 

a             b 

13. 

2x^-~bx+9  _x''-1x+b 

a—x      b —X 

2x^-1x+?>     x'-Sx+2 

7. 

a  +  x  __a  +  b 
a—x      a—b 

14. 

ax  +  b  —  e {b  —  eY 

ax-b  +  e      {b  +  ey 

15 

- 

a^x 
a'-b' 

2e        o 

= —2cx. 

a  —  b 

a^  +  ab  +  b'' 

16. 

2x-^1  _x+1 
2^-3      :?;+ll 

19. 

23:^:  + 54         36a;- 7 
115^-29      180^+23 

17. 

4:x-b  _10:?;-32 
2x+l0       bx  —  Bi 

20. 

210^-73_21:r+7.3 
310:z;-66       31:?;  +  8 

18. 

57^-43      ?>9x- 

_7 

21. 

mx—a—b  __  rax — a—c 

19:?;+ 13      13  a; +25  mx—c—d      nx—b—d 


182  LINEAR    EQUATIONS. 

x^  -\-  ax"  —  hx -\-  c  _x^  ^  ax  —  h 
x^  —  ax^  -^hx  -\-  c      x^  ■—  ax  ^  b 

gV  +  a^hx'^  —  acx  +  d  _  dx^  +  abx  --  c 
aV  —  a'^bx'^  +  acx  +  d      aV  —  abx  +  c 

8x'  +  Ux'  —  Sx  +  5  _4:X'  +  Qx  —  4: 
'    Sx'-l2x'  +  Sx  +  5       4:X'-6x  +  4:' 

be  ac  ab         \a      b      c  J 


aoc  o         a6c         \        b^ 


X. 


28.  ^         •         ^^ 


(a +  5)^  '   (a -5/      (a^-Z)^)(a  +  Z))      {a -by 

„  :g  +  g ^  —  5 . x  +  c 

{a  —  b)(c~  a)      (a  ~b){b  ~  c)      (b  —  c)(c—  a) 

a-\-  c 

{a  --b)(b  —  c)  (c  —  a) 

fx  +  2a\  ,       fa  +  2xV     ^ 

30.    x[ — ' \  A.  a  [ — ' p=2a. 

\x  —  a  J  \ a  —  X  J 

x  +  a  x  —  a  a^ 


x'^+ax  +  a^      x^  —  ax  +  d^      x  (a;*  +  a  V  +  a^) 

32     x  +  ct^C^^  +  ct  +  c^ 
'    x  +  b      \2x  +  b  +  c)' 


33 


{x+df-b''      {x+by-a^     :^-{a+by  '  x''-{a-by 

\   X+l  X+4:    y  \   X+2     ^     X+Z    J 


LINEAR    EQUATIONS.  183 

§  41.    Sometimes  a  factor  independent  of  x  can  be  dis- 
covered and  rejected. 


Examples, 
?>ahc      hx  ,      c^}? 


1. 


a-^h       a       {a  +  hf 

Transpose  —  and  factor  ;  then, 
a 


ah 

a  +  h 


Q      ,        ah 

36'-' 


{a  +  h) 
ah 


] 


"a  +  h 

x+a  x—h 


{a  —h)(c  —  a)      {a  —  h)(h  —  c)      (h  —  c)(c  —  a) 

h  +  c 

(a  —  h)(h  —  c)  (c  —  a) 

Add,  term  by  term,  the  identity  (Tb.  III.,  page  67), 

x—a . x—h . x~c         _ 

(a  -h)(c  -  a)      (a~h)(h  ~  c)      (h  -  c)  (c  -  a)~ 

.  2x  ._ h  +  c 

'  '  (a~hy(c  —  a)      {a~-h){h  —  c){c  — a) 


■Hm 


184  LINEAR    EQUATIONS. 

3.    {x  +  a  +  by  +  {a  +  by  -{x  +  bf  -  (x  +  af 

+  x^  -\- a^  -\- b^  =  abc. 
The  left-hand  member  vanishes  for  ^  =  0,  and  hence, 

by  symmetry,  for  a  =  0  and  ^  =  0 ;  therefore,  it  is  of 

the  form  Tuabx,  in  which  tji  is  numerical. 
Put  x  =  a=^b,  and  m  is  found  to  be  6. 
Hence,  the  equation  reduces  to 

6abx  ^=  abc,  and  .'.x=^^c. 


4. 


/x  —  ( 
\x  —  I 


-  aY_x  ~2a  +  b 


b)      X  —  2b  -{-a 
Let  X  —  b  =  7)1,  X  —  a  =  n,  and  hence,  7n  —  n  =  a  —  b; 

then  we  have 

n^       n  —  {m  —  n) 2n  —  m 

m^      ■m  +  (r)i  —  n)      2  m  —  n 

.*.  2mn^  —  n^=  2m^n  —  m^. 

.'.  m^  —  n^  —  2m7^ {m^  —  rf)  ^=  0. 

.*.  (7n^  —  7f)  {m}  -\-n^  —  2 mn)  =  0. 

.-.  (m  +  n) (m  —  ny  =  0. 
But  771  —  n  =  a  —  b,  and  rejecting  this  factor,  which 

does  not  contain  x, 

w,-\-n  =  0. 
But  7n-\-n^=2x  —  a  —  b. 

.'.2x-a  —  b  =  0. 

.'.x  =  ^(a+b). 

Ex.  53. 

1 .  a(b  —  x)  +  b (c  —  x)  —  b (a  ~  x)  -{-  ex. 

2.  (a  +  bx){a  —  b)  —  (ax  —  b)  =  ab(x  +  l). 

3.  (a-b)(x  —  c)  +  (a  +  b)(x  +  c)  =  2(bx  +  ad). 

4.  (a  -  b)(x  -  c)-{a  +  b) {x  +  c)  +  2a{b  +  c)  =  0. 


LINEAR   EQUATIONS.  185 

5.  {a—h){a  —  c){a'\-  x)  +  (a  +  b){a  +  c) (a  —  x)  —  0. 

6.  {a  —  h){a--c  +  x)  +  {a  +  h){a+c  —  x)  =  2a^. 

Solve  in  (x  —  c). 

7.  (m  +  a)  (a  +  ^  —  ^)  +  {<^  —  '^)  (^  —  x)  =  a(m-\-  h). 

8.  m(a  +  6  —  :r)  =  n(.:z;— a  —  ^). 

9.  (?7^  +  7z) (m  —  n  —  x) -{- r)i{x —7i)  —  n (x—m)  =  m 

10.    ^  —  ^  J  ^  — ^  ,  i^  — ^_^3 
m  n  p  ' 

a^b  ~  X      Ij^c  —  X  ,  c^a  —  x 


n\ 


a  b  c 

-  fv  €C  X     I      U  X     I      o  ^ r\ 

^T      I         ^  i — jr~^- 

DC  ca  ab 

1  ~  ax  ■   1  —  bx  ■   1  —  g^  _  pv 
be  ca  ab 

Deduce  tlie  solution  from  that  of  No.  12. 

-.     a  —  bx  ,  b  —  ex  ,  c  ~  ax      ^ 

14.    — - — + + j-=-0. 

be  ea  ab 

-  J,     /     I   7    t     \        a'^  -{-b^      2  abx  ,   fa-\-b\ 
a  ~  6       a  +  b      \a  —  bj 

16     3a5g  ■       a^b''         {2a-^b)V'x ^{b +  ?>ac)x 
a+b      {a  +  by'^    a{a  +  bY  a 

,^     10  ,  4      9  ,  2         Q  ,      .     1 

17. f--  =  -  +  -.        Solve  m  - 

X       ^      X      6  x 

18  71^23-a;7        1 

*   re      3~     3:^         12     4^* 

19  7      13__2(5:r-12)      17  ■  10 
'3      5^  Zx  %)      x' 

20  ^Q~^  j_  IB  +  rg ^  7:g  +  266  _  4:?;+17 

3      "^      7  .T  +  21    ~        21      * 


186  LINEAR    EQUATIONS. 


21.^+         3  1 


22. 


x  +  3      2(x  +  3)      2     2(x  +  3) 

6:^  +  5       1  +  8^^1  — :r     x-S 
3^-15   "      15  3  5    ' 


1  1 

a—-  X—- 

23.    ^-^^ 2-^.  24. 


,1a;  ,1a  A ^ 


X 


25.  (:^-l)(^-2)-(ri;-3)(rr-4)  =  3. 

26.  (r^-3)(:i;-4)-:(:r-2)(:i'-6). 

27.  2(^-4)(3:r  +  4)  +  (2:^-3)(3^  +  2) 

-6(^-2)(2:^-3)=.0. 

28.  (a  —  :r)  (Z)  —  a;)  ==:  2:1  30.   {a  —  x){h-{-x)=^h''  —  x\ 

29.  ((2  — :?;)(rr— 5)^ri;'  — c'.    31.   (x  —  a){x  —  h)=zx''  —  o?, 

32.  (a  +  rr)  (5  +  r^:)  =  (a  —  rz;)  (h  —  x). 

33.  (a^  +  ^)  (Z):^  +  a)  =  (5  —  a:?;)  (a  —  Z^rr). 

34.  (<2  —  ^)  (Z>  —  :r)  +  (a  —  c  —  a;)  (r^  —  Z)  +  (?)  =  0. 

35.  (a  —  r?;) (Z>  —  x)  —  {c  —  x){d  —  x)  =  (c  +  d)x  —  cd. 

36.  (:x;  --a)(x  —  h)  —  {x-  c)(x  —  d)  =  {d  —  a) (d—b). 

37.  [(a''-b')x-ah][a-(a  +  b)x]  +  2a¥x 

=  [(a  +  byx  +  ab][b  —  (a-b)x]. 

38.  (:^  +  l)(ri;  +  2)(:^  +  3)  =  (^-3)(:2;  +  4)(r^  +  5). 

39.  (x  +  l)(x  +  2)(x  +  S) 

=  (x-l)(x--2)(x-S)  +  3(x+l)(4:X  +  iy 

40.  (:^;+l)(:^:  +  4)(:^+7)  =  (a;  +  2)(:^  +  5)^ 

41.  (x+2)ix  +  by  =  {x  +  3)\x  +  6). 

42.  (2;  -  l)(:r  --  4:)(x-6)  -x(x  -2)(x-9)-=  136. 


LINEAR   EQUATIONS. 


187 


43. 

44. 
45: 

46. 

47. 

48. 
49. 

50. 
51. 

52. 

53. 
54. 
55. 
56. 
57. 


(a  +  x){b+  x)  (c+x)  —  (a~x)(b  —  x)  (c  -  x) 
=  2(x'  +  abc). 

(x  —  a)(x—h)(x—c)  —  (d—a)(d—b)(d—c)__ 


:-d 


(x~dy. 


x{x  —  ay  —  {x  —  a  +  b)(x  —  a  +  c){x~b  —  c) 
^{o}  +  bc){b  +  c). 

{x  -  a  +  b){x  —  b  +  c){x  -  c  +  d)  -  x^(x ~-a  +  d) 
=  bc(d—  a). 

(x  —  a-{-b)(x  —  b-{-c)(x  —  c-\-d) 

—  x(x~a  +  c)(x  —  c-j-d)  =bc(d—d). 

(x~2a)  (x-2b)  (x-2c)  -  (x-a~b)  (x-b-c)  (x~c~a) 
=  (a+b+c)  (a'W+c')-9abc. 

x^~(x-~a-{~b)(x~b-]-c)(x  —  c-\~a) 

=  (a-i^b-{-c)la'+b'+c')~2(a'b  +  b'c+c'a)~Sabc. 

\        xj\         x]\        x)      X^  X 

{x  +  a)(x-\-b)-^{x^c){x-^a)  =  {x'^b){x-\-d) 
J^{x-^d){x-\-c), 

(ax  +  b)  (ax  —  c)  ~  a(b  —  x)  (ax  +  b) 
=^a^(x  —  c)(x  —  b)  —  a  (ax  ~  c)  (c  —  x). 

2x-?>     ?>x-2_bx''~2^x  —  4: 
x-4:        x-^       ^'^-12:^;+  32* 

bx-l         ?>x  +  2  _x^  —  ?>0x  +  2 
3(:^+l)      2(x~\)  6^2-6 

3:r-7      3(a:+l)_       ll:r  +  3 


2^-9      2(x  +  ^)      2x^-Zx-2i 
10r?;+7      _g 


7rr  — 5  ,  8:?;  — 7 


?>x-2     ^x-\      '^x'-'dx-\'2 

2x+1      3:r-6      b(x-l)_^x-2  ■    bx-S      2^+2 
?>x-1~^2x~b       9:1—25       2:i— 5      9a;-25      3rr-7' 


188  LINEAK   EQUATIONS. 


58. 


59. 


60. 


61. 


62. 


63. 


4:X~S  3  4:X'+2 


66. 


67. 


68. 


1  +  x 

l~x      x'  - 

X—  a 

x-h 

x  —  m 

x  —  n 

'-^ 

1             00             I 

h-' 

4     l-2x     4 
4 

a  .      ex     _<?   1      CLX 
c     ax  —  h      a      ex  —Jb 

3_1     2_1       3_2 

2 X     3 £__2 3_ 

2      .1'      ox      2x 

2(2; -1)      rg  +  8^3(5.a;+16) 

"'  2'--  7        x-4:'~     5:^-28 


7nx  —  p      nx  —  q      m      n 

-_      ax -\- h    .  ex  -\-  d      a    .  e 
b&. = 

mx  —  p      nx  —  q      7)i      n 


b  —  xc  —  x_a(s—2x) 
a  +  x      a  —  X        d^  —'  o(? 

a-\-'b      h-\-e  _a  +  e-^2h 
x  —  ax  —  h  X  —  c 

ax  ■\-h         hx     __     ax         {ax^  —  2h)h 
ax  —  h      ax-\-b      ax  —  b         aV  —  b^ 

ax  —  b  j_(^x  —  d      (bn-\-dm)x-—(bq-{-dp)_  o,    ,c 
mx—p      nx  —  q  {mx—p){nx — q)  m     n 


LINEAR   EQUATIONS.  189 


70        "m.  n  2^     ^    m  n  p 


\ 


X  —  a      X  —  b      X  ~  c      X  —  c      X  —  a      x  —  h 

ax  —  2a_ax  —  2h  2x'^  —  Sx  -}-  b  _2 

'ax  —  2b      ax  +  2a  '    7x''  —  4:X  +  2~1 

1_1         _1 

a      X  X  ryr^^     ax  —  bx-\-  c  __ct 

\      \  ~         \  map'  —nx-\~p      m 

-+-      a+- 
a      X  X 


^        ax^  —  bx^  +  ax  —  d  _  ax  —  b 
^  mx^  —  nx^  +  mx  —  q      vix  —  n 

76     4~         1         X 1      ^^    ^"""3      2      2      '^ 3 

'    1  ,      "^4      1  ,  4  '2  3    ^3  2' 

21  71  21  71 

<  O. 


o;  — 98      x—9^      x  +  4:4:     x ~  52 


80. 


.T  —  6      ;r  — 11      x-~7      x  —  \2 
9  9  2  2 


a; -51      a;— 15      x~^l      x  +  ^l 


81.    _^+_^=^_+.     1 


x  —  Q      x  —  9      x—1      x—10 

82.    J_  +  ^_=_A_  +  ^_. 
a:  —  6      a;~3      :?;  —  2      ir  —  5 

m  —  n       a  —  b  7n—-n      a  —  b 

x~a      X  —  m       x~b       x — n 

a-\-b  __  a+c b+d c-{-d 

x~-b      x-c~  x—{a+b+2c+d)      x-(cc+2b+c+d) 


190  LINEAR   EQUATIONS. 

85.    {x  +  a  +  hy-{x+ay-{x  +  by+x'-{a+by+a'+h' 
=  l2ab[x'  +  {a  +  hY]. 

o^      a  —  X    ,    b  —  x    ,    c~x  ^x 


I 


■be      h^  —  ca      &  —  ab      ab-\-bc  +  ca 

g^^    {m  —  n){x  —  a)  .  (n—p)(x  —  b)      (p—m)(x~c)^Q 
b-\-c  c-{-a  a-\-b 

88.    {x+a  +  by-{a+bf-{x  +  bf-{x+ay+x''+a'+b'' 
=  10abx(2x  +  a  +  b){x  +  a  +  b). 

ftQ       <^^  —  1     ,     bx  —1     ■      cx—1    Sx 

a' (c  +  b)      b\c  +  a)      c'(a  +  b)~  ab  +  bc  +  ca 

90       ^  —  2a     ,     X  —  2b     ,     x  —  2c    __  o 


91. 


b-{-c  —  a      c-j-a  —  b      a-\-b  —  c 

x  —  2a     ■     X  —  2b     ■     x  —  2c    _       3rg 
b -\- c  —  a      c-{-a  —  b      a-{-b  —  c      a-{-b-\-c 

Q^      a  —  X    .    5  —  ry    .     c  —  X  3 

a^  —  be      h^  -be      c^  —  ab      a-\-b  -\-  c 

^„      x-{-2ab    ,    2ab  —  x   _x—2ab    ,    :z:  +  2a6 

V/o. 1 j • 

a-\-b-{-c      b-\-c  —  a       a  —  b-^c      a-\-b  —  c 
^.  a  ,  b  a  —  c  ,  b  —  c 


96. 


x-[-b  —  e      X  -\-  a  —  c      x  -\-b      x-{-a 

171^ {a—  b)      n^{b  —  e)  .p^{c  —  d) 
X  —  m  X  —  n  x  —p 

q^ [pd -{'(n—p)e-{-  (in  —  n)b  —  Tna] _  ^ 
X  —  q 

(x-2)  (x-6)  (x-6)  (x-9)  +  (a+2)  (a-4)  (a-5)  (a-11) 

X 

(h+l)  (1+5)  (b+8)  (b+U)  ^  ^^_^^  ^^_^^  (^„,,) 

X 

,  (a'-l)(a-8) (a-10)+(6+2) (b+3) (5+10) (&+1 1) 


linear  equations.  191 

Equations  Kesolvable  into  Linear  Equations. 

§  42.  In  order  that  the  product  of  two  or  more  factors 
may  vanish,  it  is  necessary,  and  it  is  sufficient,  that  one  of 
the  factors  should  vanish.  Thus,  in  order  that  {x—d){x — h) 
may  vanish,  either  x — a  must  vanish,  or  x  —  h  must  vanish, 
and  it  is  sufficient  that  one  of  them  should  do  so. 

Hence,  the  single  equation  {x  —  a)  (^  —  ^)  =  0  is  really 
equivalent  to  the  two  disjunctive  equations, 

x  —  a  =  0  or  x--h^=0, 

for  either  of  these  will  fulfil  the  conditions  of  the  given 
equation,  and  that  is  all  that  is  required. 

Similarly,  were  it  required  to  find  what  values  of  x  would 
make  the  product  {x—a)(x—h)(x—c)  vanish,  they  would 
be  given  by 

X  —  a^=0,  or  rr  —  5  =  0,  ov  x  —  c=^0. 

.'.  x  =  a  or  h  or  c. 

Hence,  the  single  equation  (x—a)(x—b)(x—c)  =  0  is 
equivalent  to  the  three  disjunctive  equations 

x  —  a=:0,  ov  x  —  b=^0,  or  X  —  c=^0. 

Examples. 

1.  Solve  ^' —  :r  -  20 -=  0. 

The  expression  =  (:z;— 5) (r?;  + 4),  which  will  vanish  if 
either  of  its  factors  does  ;  that  is,  if  ^~  5  —  0,  or  :i;  +  4==0. 
.'.X  —  5,  or  x  =  —4. 

2.  Solve  x'~x^  —  x''  +  x  =  0. 

This  gives  x^{x— 1)  —  x(x  —  1)=^  x(x —1) {x^  —  1) 
=  x{x-l){x+l){x-l), 
which  vanishes  for  a;  =  0,  rr  =  1,  x=^  —  1. 


192  LINEAR   EQUATIONS. 

3.  Solve  x\a-b)  +  d'(b  -  x)  +  h\x  -  a)  =  0. 
.-.  x\a  -b)—x{o?-  b')  +  ab{a-b)^  0. 

.-.  {x  ~a){x-  b)  {a  —  b)  =  0. 

If  a— Z>  =  0,  tlie  given  equation  will  hold  irrespective 

of  the  values  of  ^— a  and  x—b,  and  therefore  of  the 

values  of  x ;  but  \i  a  —  b  be  not  zero,  then  must  either 

X  —  a^=0,  ov  x~~b  ^=^0. 

.'.  x^=  a,  or  X  =  b. 

4.  Solve  221^^-5^-6=.0. 

Here  we  have  the  factors  17:?:  —  3  and  13  a:  +  2 ; 
hence,  the  equation  is  satisfied  by 
17^:  — 3  =  0,   or  x^^, 
and  also  by  13rr  +  2  =  0,  or  x  =  —  ■^^. 


5.    ^o\YQ{x-af  +  {a-by  +  {b~xy  =  0. 

The  expression  is  equal  to  2>(x  ~  a)  (a  —  b')(b  —  x),  and 
therefore  vanishes  for  x  —  a  =  0,  or  x=^  a;  and  for 
X  —  b  =^0,  or  x  =  b. 

Ex.  54. 

1.  If  an  equation  in  x  have  the  factors  2r?:— 4  and  2.r  — 6, 

find  the  corresponding  values  of  x. 

2.  If  an  equation   give  the   factors  2x  —  \   and   3:?:— 1, 

what  are  the  corresponding  values  of  :r  ? 

3.  If  an  equation  give  the  factors  ^x^~12  and  4.r  — 5, 

find  the  corresponding  values  of  x. 

Find  the  values  of  x  for  which  the  following  expressions 
will  vanish  : 

4.  x''-2x+l]   4:x''--l2x  +  9. 

5.  9x^-4:]    x'~(a  +  by;  x''~2ax  +  a\ 


LINEAR    EQUATIONS.  193 

6.  x'-9x  +  20]   4^' -18:?; +  20. 

7.  x'  +  x-6;   x^-x-Vl',    9:r^  -  9a;- 28. 

8.  6^-^-12:r+6;    6:r^-13^'+6;  6:r^-20:r  +  6. 

9.  ^x^-~hx-^\    6:?;^-37^  +  6;    6^^  +  :?;  — 12. 

10.  A  certain  equation  of  the  fourth  degree  gives  the  fac- 

tors x^—x  —  2i  and  4 a;^ — 2  a; — 2.    Find  all  the  values 
of  rr. 

Find  the  values  of  x  in  the  following  cases  : 

11.  x^~'2.'bx^-Z'b''x=^{). 

12.  x^  ~  ax^  —  o?x-\- a^  ^=^^. 

13.  x''-Zx-\-'l=^^. 

14.  a;'  — 2a^  +  2a'^  — a^  =  0. 

15.  x^-\-(h  +  c)x'-hcx-})'c-hc^  =  ^. 

X  —  a  j^x  —  h  ^        (a  —  b)'^        x^  —  a^ 

X  —  b      X  —  a      (x  —  a)(x  —  b)      {x  —  a){x  —  b) 

17 .  x^  -  bx"  -  a^x  +  a^b  =  0. 

18.  Zx^'  +  babx'-^ a^y'x  -  4 aW  =  0. 

19.  x\a~b)  +  a'(b-x)  +  b\x-a)  =  0. 

20.  (^-^)(^-g)   I  (x-c)(x-a)^-^^ 
(a  —b)(a  —  c)      {b  —  c)(b  —  a) 


fx--2aW 

X    — +  a 

\x  +  a  J 


"Ax  — a 


21.  x[^     ■        l  +  af        ,        \=^x^-a\ 
x-\-  a  J 

22 .  {x  +  a  +  bf  -  x^  -  o?  -b^  =^  {x  +  a)  (a^  -  b""). 
2o  ctb .  bx  ,  ax 


(b—a)  (x—  a)      (x—a)  (a—b)      (a—b)  (b—x)      a—b 
24.    Form  the  polynome  which  will  vanish  for  a;  =  5  or  —  6 


194  LINEAR    EQUATIONS. 

25.  Form  tlie  polynome  which  will  vanish  for  x^=a  or  4a 

or  3(2  or  —4  a. 

26.  Form  the  equation  whose  roots  are  0,  1,  —2,  and  4. 

§  43.  Employing  the  language  of  Algebra,  the  principle 
illustrated  in  the  preceding  section  may  be  stated  as  fol- 
lows : 

Definition.  Any  quantity  which  substituted  for  x  makes 
the  expression  fix)  vanish,  is  said  to  be  a  root  of  the  equa- 
tion f{x)  =  0.  Thus,  if  a  be  a  root  of  the  equation /(:?;)  =  0, 
then /(a)  =  0. 

By  Th.  I.,  if  a;  — a  is  a  factor  of  \h.Q  polynome  f  {xY ,  then 
f{ay  =  0,  and  a  must  be  a  root  of  the  equation  f(xy  =  0  ; 
hence,  in  solving  the  equation,  we  are  merely  finding  a 
value,  or  values,  of  x,  which  will  make  the  corresponding 
polynome  vanish.  Suppose  f(xy  =(x  —  a)<j>  {xY~^  =  0,  we 
are  required  to  find  a  value,  or  values,  of  x  which  will  make 
(x~  a)cf>  (xy~^  vanish.  The  polynome  will  certainly  vanish 
if  one  of  its  factors  vanishes,  whether  the  other  does  or  not, 
and  will  not  vanish  unless  at  least  o?ie  of  its  factors  vanishes. 
Hence,  (x  —  a)<f>{xY~^  will  vanish  if  x  —  a  =  0,  quite  irre- 
spective of  the  value  of  <^  (5;)"~\  Also,  if  c^  {xY''^  =  0,  the 
polynome  will  vanish,  irrespective  of  the  value  of  x  —  a. 
It  follows,  therefore,  that  iif{xY  can  be  resolved  into  two 
or  more  factors,  each  of  these  factors  equated  to  zero  will 
give  one  or  more  roots  of  the  equation /(:r)**  =  0. 

"When  there  can  be  found  two  or  more  values  of  x  which 
satisfy  the  conditions  of  given  equations,  they  are  some- 
times distinguished  thus  :  Xi^  x^^  ^3,  etc.,  to  read  "one  value 
of  :r,"  "a  second  value  of  ^,"  ''a  third  value  of  a;,"  etc. 
Thus,  if  ^  ^^  _  ^^  ^^  _  ^)  (^  _  ^)  _  Q^ 

.*.  Xy  =  a,    ^2  —  ^,    ^3  ==  c. 


LINEAR    EQUATIONS. 


195 


Examples. 
Solve  2x^  -  l^x"  +  21 X-  18  =  0. 
Factoring,  (^x  -  2)  {x  —  ?>){2x-^)  =  0. 

,   .  X^  Zi^     X2  ' — ■  Oj     ^3  ■^^* 

x"^  —  (a  +  Z>)  a;  +  (0^  +  ^)  ^  =  (<3^  +  <^)  ^. 

.-.  x'~(a  +  h)x  +  (a  +  c)(b-c)==  0. 

.  • .  ^r^  -  [(a  +  c)  +  (5  -  c)]  a;  +  (a  +  c)  (^>  -  c)  =  0. 

.-.  [:r  -  (a  +  c)]  [^  -  (^  -  c)]  =  0. 

.*.  :z:i  ==  a  +  <?,  X2  =  h  ~  c. 


(a'  +  b')x-(a''~b'). 


{a^-b'')x-{p?  +  b^) 
.  ^-M^aH^-l) 
"x-l      b'^x+l) 
,  Xi  +  1      a 


1=0.        .-.xr- 

iPg  —  1      c) 

(g  -  xy  +  (b  —  xf 


V^  -  ly    5^ 


a~b 
a  —  b 
a  +  b 


_34 

{a  -  xf  +  (a  -  rr)  ib-x)  +  (b-  x)"      49* 


{a-xy-2{a-x){b-x)+(b-xf      3(34)-2(49) 
r(a-a;)  +  (^-a;)-l^ 

a  —  6> 

a  — 5 

(3;  —  a){x~  b)  ,  (g;  —  5)  (:g  —  c)  __  -j 
(c  —  a)  (c  —  b)      {a  -~b)(a  —  c) 


16. 


196  LINEAR    EQUATIONS. 

Subtract  term  by  term  from  the  identity  (see  page  67) 

(x  —  a)(x  —  h)  .   (x  —  h)(x  —  c)   ,   (x  —  c)(x  —  a)  _  -. 
(c  —  a)  (c  -  b)      {a  —  b){a  —  c)      (b  —  c)  (b  ~  a) 

:.{x~c){x  — a)=^0.     :.Xi^=c,   2^2  =  a. 

6.  Find  the  rational  roots  of  x^— 12:^^+51^'^— 90:?;  +  56  ==  0. 
Factoring  the  left-hand  member  by  the  method  of  §  27, 

{x  -  2)  (2;  -  4)  (;r'  —  6^  +  7)  =-  0. 
:.Xi  =  2,    ^2  =  4,    or  ^^  —  6^  +  7^0. 

Since  x^  —  ^x  +  1  cannot  be  resolved  into  rational  fac- 
tors, we  know  that  it  will  not  give  rational  roots ; 
therefore,  ^1  =  2,  a^g  =  4  are  the  only  values  that 
meet  the  condition  of  the  problem. 

In  order  that  two  expressions  having  a  common  factor 
may  be  equal,  it  is  necessary  either  that  the  common  factor 
should  vanish,  or  else  that  the  product  of  the  remaining 
factors  of  one  of  the  expressions  should  be  equal  to  the 
product  of  the  remaining  factors  of  the  other  expression, 
and  it  is  sufficient  if  one  of  these  conditions  be  fulfilled.  In 
symbols  this  is 

If  {x  —  a)f(x)  —  (x~a)(f> {x),    r.Xi  —  a  OTf(x)  =  <^ (x). 

7.  ^+-  =  a^ — 

X  a 

1      1         X  —  a      X  —  a 
:.x  —  a 


ax  1  ax 

.'.  x~— a  =  0,  or  ax=^\.     /.Xi  =  a,   X2--=  — 

a 

(x  +  a+bXx+b  +  c)  =  (x  —  Sa  +  b)(2x-Sa  +  2h-c). 
.    x  +  a  +  b  _  2x~da  +  2b  —  c  _  x  —  4:a+b  —  c 
x  —  Sa-\-b  x-}'b-{-c  Sa-\-c 

Page  155,  (5). 


LINEAR    EQUATIONS.  197 

2(x  —  a-\-b)  __x  —  a-\-  b 
X  —  3a  +  b  3a  +  (? 

.'.  Xi=^  a  —  b. 

i(^2  — 3a  +  5)  — 3(2  +  ^.     :.  X2=^9a  —  b  -\-2c. 

9.  {x~2)  (x-b)  (x-6)  (x-9)  +  (y+2)  (y-4)  (y-5)  (y-11) 

+  (z+l)(z+5)(z+8)(z+12) 

=  x(x-A)  (x-7)  (^-ll)  +  (y+l)  (y-1)  (y-8)  (y-10) 

+  (.+2)  (.+3)  (.+10)  (.+11). 

Let:?;'-=:i;'— ll:r,  y'=f-97/,  and  .'==.'  +  13.. 
.•.(^'+18)(:r'+30)  +  (y'-22)(y'+20)  +  (.'+12)(.'+40) 

-  x'  (x'+  28)  +  (y '-10)  (y'+  8)  +  (.'+  22)  (.'+  30). 
.•.^^''  +  48:r'+540+y'*'^-2y'-440  +  .'2  +  52.'+480 

=::2;''+28rr'  +  y'^-2y'-80+.'^+52.'  +  660. 
.-.  20:?:'-0.  :i;'-lla;=-0,  ^^i  =  0,  rr.^ll. 

Ex.  55. 

What  can  you  deduce  from  the  following  statements  ? 

1.  ^  .  ^-0.  3.    (a-b)x  =  0. 

2,  A  '  B  '  C=0.  4.    12^y==0. 

5.  What  is  the  difference  between  the  equation 

(:^;-5y)(:r-4y+3)  =  0 
and  the  simultaneous  equations 

a;  — 5?/  =  0  and:^;  — 4?/  +  3  czr  0? 

What  values  of  x  will  satisfy  the  following  equations  ? 

6.  x(x—a)  =  0.  11.    x(x'  —  a'')=^0. 

7.  ax(x  +  b)  =  0.  12.    a'x^^b'^x. 

8.  (x~a)(bx-  c)=^0.         13.    x' +  (a  —  xf  =- a\ 

9.  ax^  =  3ax.  14.    ^' +  (a  — 2;)'  =  (a -2^)1 

10.  x'  =  (a-\-b)x.  15.    (a-:r)'+(:z;-^)'-:a'+6l 


198  LINEAR    EQUATIONS. 

16.  (a  —  x)(x  —  b)  +  ab  =  0. 

17.  (a - xf  --(a-x)(x-b)  +  (x-  bf  =  a' +  ab  +  b\ 

18.  x^  —  {a-b)x  —  ab^^. 

19.  x^-~{a-\-b-\-c) x^  +  {ab  +  be  +  ca) x  —  abc  =  0. 

If  X  must  be  positive,  what  value,  or  values,  of  x  will 
satisfy  the  following  equations : 

20.  (x-5)(x  +  4:)  =  0.  23.    3:^2-10r?;  +  3-:0. 

21.  :?;'  +  29:?;-30==0.  24.    x' -ISx^ +  ^6  =  0. 

22.  x'-17x-84:-=0.  25.    :i;'-2^'  — 5a;+6  =  0. 
Solve  the  following  equations : 

26.  (a~xy  +  (x-by  =  (a-by. 

27.  (a  —  xy  -  {a-  x){x  —  b)  +  {x  -  by  =  (a-  by. 

28.  a'(a~xy  =  b'(b-xy.     29.    a'(b-xy  =  b\a—xy, 

30.  (:r-a)^  +  (a-^>)'  +  (^-:r/-:0. 

31.  {x~iy  =  a(x'-l). 

r.n     a  —  x x  —  a  ^„     a-\'b  —  x      a  —  c  +  x 

X  —  b      c -\- X  a~c  —  X      a-\-c  —  x 

34.  {x  —  a  +  b){x  —  a  +  c)  =  (a— by  —  x\ 

35.  (x-ay  —  b''  +  (a  +  b-x)(b  +  c~x)  =  0. 

36.  (a  +  b  +  c)x''~(2a  +  b  +  c)x  +  a=0. 

^^     a  +  b~x_a~]-b  —  c 
c  X 

38.  (a~xy  +  (a~by=(a  +  b-2xy, 

39.  a;(a  +  ^>— :r)  +  (a  +  Z>  +  c)c  =  0. 

40.  (n—p)x'^-\'(p~7ri)x-]'m  —  n^=0. 

,-       aa;^  — 5:r  +  c        c?  --,       ax"^  —  bx4-c        a~b -{- c 

41. ! = -.         42.    ! —  = — 

rnx  — nx-{-p     p  mx^  —  nx-\'p      m — n-\-p 


LINEAR    EQUATIONS.  199 

43.  ^x^  +  a'-h''-2{a  +  h)x={a~x){h+x)~{a+x){h-x). 

44.  (2a -h-  xy  +  ^{a~  by  =  (a  +  b~  2x)\ 

45.  {2a  +  2c  —  xy  =  {2h  +  x){?>a-h  +  ?>c-2x). 

46.  {^a-bh  +  x){pa-?>h-x)^(la-b-^x)\ 

47.  {2>a-h  +  x){Za  +  h-x)  =  (ba  +  U~Zx)\ 

48 .  a{a  —  h)-~b{a  —  c)x  +  c(h  —  G)x^=^0. 

49.  {ab+bc  +  ca){a:'  +  x+l)  +  {a-by 

=  (2ac  +  b')(x'  +  x+l)  +  (a-cfx. 

50.  (x+l)(x  +  S)(x-4:)(x-1) 

+  (x-l)(x-S)(x  +  4:)(x  +  1)  =  96. 

51.  (x-l){x  +  S)(x-b)(x  +  9) 

+  (x+l)(x-3)(x  +  5)(x-9)  +  18  =  0. 

52.  x  +  l  =  3i.  56.    ^-^.^-^      IB 


X  X 


—  b      a  —  X       6 


,1      a-{-b  ,  a  —  b         ^^     a—x      b-{-x      m      7i 
53.    x-\ — = — '—•-I -•         57.  '     — 


X      a  —  b      a-{-b  b-\-x      a—x      n      m 

lab  ^^     a  ,  X      m  ,    n 

54.    X =  -  — -•  58.    _-(-_=: 1 

X      b      a  X      a      n      m 

55^    a+_^  ,   Mif^2i  59.    (^-^)'  +  (^^-^)'^^ 
b -\- X      a  +  x        ^'  {a—x){x—b)         2 


60. 


61. 


62. 


63. 


{x + ay  +  {x  -by  _^d'  +  b'' 

{x  +  ay-lx-by        2ab  ' 

(a-xy-(x-by_    4:ab 
(a  —  x)(x  —  b)         a^  —  b"^ 

(a  —  xf  +  {a  —  x){x -b)  +  {x -  by __4:9 
(a  -  xy  -la-x){x-b)  +  (x  -  by      19* 

2a'  +  a(a-  x)  +  (a  +  ^)'  _  3 
2a'  +  a(a  +  x)  +  (a-xy      2 


200  LINEAR    EQUATIONS. 

64.  (b--xy  +  (2-xy  =  i7. 

65.  (x-ay  +  (a-by  +  (b-xy  =  x'-a\ 

66.  (a-xy  +  (x~hy--=(a-by. 

67.  (x  +  ay  -  (a  +  by  +  (b~  xy  =(x  +  a)(x  +  b)  {a  +  b), 

68.  x^-{x~by-{x-a  +  by  -  cc"  +  {x~  ay  +  (a  -  by 

+  b^  =  {a-b)c\ 

69.  {x  +  ay  -{x  +  by  -(x-  by  ~  (2ay  +(x-  ay 

+  (a  +  by  +  (a~ by  -  (a'  -  b')c. 

70.  (x-a+by~(x~ay  +  (x~by~x'  +  a'-(a-by:^-b\ 

^-     (a  —  xy  +  (x  —  byo^^/         7  X 

73.  2(a-  xy  -  9 (a  -  :z;)^(:^^ -  ^>)  +  14 (a  -  xy(x  -  5)^ 

-  9(a  -  ^)(:r  -  ^>y  +  2(x  -  Z>/  =  0. 

74.  4(6^  -  :?;)*  -  17(a  -  :r)^(^  -  ^)^  +  4(:z;-  Z))*  =  0. 

Find  tlie  rational  roots  of  the  following  equations : 

75.  a;*-12^^  +  49a;2-78^  +  40---:0.     Letz^x'-Qx, 

76.  x'-6x'  +  7x'  +  6x~8  =  0, 

77.  :r*  -  10:r'  + 35a:' -50:^;  + 24  =  0. 

78.  32:r'-48:2;'-10a;'  +  21^+5  =  0. 

79.  ^.-3- 6:1;'  + 5a: +  12  =  0. 

80.  lla:^  +  10a;^-40a;  =  176. 

81.  5 4 9 4        ^        5      _Q^ 

X     X  —  a     x--2a     x  ~Sa     x  —  4:a 


/?_ 


LINEAR    EQUATIONS.  201 


82.       14      ,      5  4     __     14 


:r  +  20      x+b      x  —  4:      a;  — 55      x  —  AO     x-2b 

g       2x+5a      x-\-8a  .       x      _  x  —  a     ^4- 5a  ,  2x—ba 
X  x—a      x  —  2a      x  —  '^a      x~4:a      x~ba 

84     ^-^4  ■  x  +  2  .  x  +  4:_^x  +  ^      x-1      x  —  S 
x-]-2         X         X  ~1      x  —  2     x—3     X  —  5 

X      x  —  1      x—2     x~-3     x  —  4:      X'-5     x  —  6 

86.  T' -  f"^^]  x'  -  ^2^^'-  =  X + ^« -  ^^  ^1  -  '■^' 


a  +  bj  1-j-  ex  V^  +  ^J  \1  +  ^^- 

g^^    4:x'  +  4:a'-33xV  ^  ,  .^^,  _  ^^,^  +  9^a2  _  2a'\ 
2x  +  a  ^  ^ 

88.  ^  •  ^  -  ^*^ 


89. 


:r*^  -  11  :r  +  28      x'  -  17.T  +  70      x'  -  14^'  +  40 

8  ■  8  ^  ^^ 

x'  —  Qx  +  5  ^'  -  14a^  +  45      x'  —  10:r  +  9* 


90.    Q(a~  xy  -  25 (a  -  xf(x -h)  +  38(a -  xf^x  -  by 
-  25(a  -  n^)(:^;  -by  +  6(x~  by  =  0. 


CHAPTER  VI. 

Simultaneous  Linear  Equations. 

§  44.  There  are  three  general  methods  of  resolving 
simultaneous  linear  equations  :  first,  by  substitution  ; 
second,  by  comparison  ;  third,  by  elimination.  The  last 
is  often  subdivided  into  the  method  by  cross-multipliers, 
and  the  method  by  arbitrary  multipliers. 

In  applying  the  elimination  method,  the  work  should  be 

done  with  detached  coefficients,  each  equation  should  be 

numbered,  and  a  register  of  the  operations  performed  should 

be  kept. 

Ex.  Resolve  u  +  v  +  x  -\-  y  +  2=  15, 
U  +  2V+  4:X+  8y  +  16z=  57, 
U  +  SV+  9a;  +  27  y  +  8X2:=  179, 
u  +  4:v  +  16x+  64y +  256z  =  453, 
u  +  5v-\-26x+  1252/  +  6252  =  975. 

u      V      X  y  z 

Register.                                       1      1       1  1  1  =    15  (1) 

12      4  8  16=    57  (2) 

1      3      9  27  81  =  179  (3) 

1      4     16  64  256  =  453  (4) 

1       5     25     125  625  =  975  (5) 

(2)-(l) 13  7  15=    42  (6) 

(3) -(2) 1       5  19  65  =  122  (7) 

(4) -(3) 1       7  37  175  =  274  (8) 

(5)  -  (4) 1       9  61  369  =  522  (9) 

7) -(6) .      2  12  50=    80  (10) 

8) -(7) 2  18  110  =  152  (11) 

(9) -(8) 2  24  194  =  248  (12) 

11) -(10) 6  60=    72  (13) 

12)-(11) 6  84=    96  (14) 

(14  -(13) 24=    24  (15) 

it(13)- 60(16)] 1           =2  (17) 

H(10)- [12(17) +  50(16)]} 1  =3  (18) 

(6) -[3(18) +  7(17) +  15(16)]     .     .     .1  =4  (19) 

(D-  (19) +  (18) +  (17) +  (16)]   .1  =5  (20) 


SIMULTANEOUS    LINEAR    EQUATIONS. 


203 


An  examination  of  the  Register  will  show  how  easy  it 
would  have  been  to  shorten  the  process ;  thus,  (10)  is 
(7)  -  (6),  which  is  (3)  +  (1)  -  2(2)  ;  similarly,  (11)  is 
(4)  +  (2)  -  2  (3)  ;  therefore,  (13)  is  (4)  +  3  (2)  -  3 (3)  -  (1), 
etc. 


Ex.  56, 


Solve  the  follow^ing  systems  of  equations  : 


1. 

2:,^  +  3y  =  41, 

4. 

i«-¥2/  =  l. 

3:t-f  2y=-39. 

|.r-|y  +  5  =  0 

2. 

5^+7y=17, 

5. 

iy  =  i«-i. 

7x~5y=    9. 

\y  =  ^x-\. 

3. 

ix  +  y  =  e>, 

6. 

l.bx-2y  =  l, 

3.T-4y  =  4. 

2.5a; -3?/ =  6. 

7.    3.5x  +  2^y  = 

=  13  +  4 

\x-?..by, 

2|rr  +  0.83/  = 

=  22i  +  0.7^-3|y. 

8. 

1  ,  1_5 

X      y      6 

11. 

y 

X      y      6 

Ux     *^-4  =  2. 

y 

9. 

X     y        ' 

12. 

8  +  y     *- 

15      4_ 
X       y"" 

6      y        ^ 

0. 

1.6     2.7 

x~  y     '■ 

13. 

5a;     0.3_g 

0.7^  y 

0.8  ,  3.6      . 

=5. 

X        y 

10x^9_3j 

7      y 

204  SIMULTANEOUS    LINEAB,   EQUATIONS. 

14.    ^x-i(y+l)  =  l,  j^     ^  +  2y  +  l_o 

i(^+l)+f(y-l)  =  9.        ■    f-^+] 

5  7  «-y+3 


15. 


x  +  2y     2x  +  y  ^^         a;  +  3y+13      _^q 

7      _     5  '    0.4a; +  0.5y- 2.5 

3a;_2~6^'  0.8a:  +  0.1  y  + 0.6  _1 

5a;+3y-23         2' 

16.    il±l^  =  8  19     ^+1      .y+2_2(a;-,y) 

«~y         '  "345 

3y-5  4  3^ 

2a;-y  +  3      a:-2y  +  3_, 

3  4 

3ar-4,y  +  3  ,  4a:-2,y-9_. 

4  "^  3  ■ 

21.  20(a;+l)  =  15(y+l)  =  12(a:  +  2/). 

22.  (a:-2):(y  +  l):(a;  +  2/-3)::3:4:5. 
,    23.    (a;-5):(2/  +  9):(a:  +  2/  +  4)::l:2:3. 

24.  ^+§=^±8, 
a;+l     3/+5 

2a;-3   __bx-& 
2(y+l)      52/+ 7 

25.  (^-4)(2/  +  7)  =  (a;-3)(y  +  4), 
(a;  +  5)(y-2)  =  (a;  +  2)(y-l). 

26.  (a;-l)(52/-3)  =  3(3a;+l), 
(a;-l)(4y  +  3)  =  3(7a;-l). 

27.  (a;+l)(2y+l)  =  5a;+    9y+l, 
(a;  +  2)  (By  +  1)  =  9a:  +  13y  +  2. 


SIMULTANEOUS    LINEAR    EQUATIONS.  205 

28.    (3^-2)(5y+l)  =  (5:t'-l)(3/  +  2), 
(3^-l)(y  +  5)  =  (a;  +  5)(73/-l). 

29.  x  +  y  =  2>l,  37.    x  +  y  +  z  =  ?>, 

3/  +  2  =  25,  2^.+  4y  +  8^  =  13, 

2;  +  a;  =  22.  3:r  +  9^  +  27 ^  =-  34. 

30.  2x  +  2y=1,  38.  :r  +  2y  +  32;  =  32, 
7:r  +  9^  ==  29,  2:r  +  3y  +  z  =  42, 
y  +  d>z=Vl.  ?>x  +  y  +  2z-=M). 

31.  l.?>x~\.9y-=^l,  39.  :r  +  y+22  =  84, 
1.7y-l.lz-=2,  x  +  2y  +  z  =  33, 
2.92;  -  2.1  :i:  -  3.  2a;  +  3/  +  0  ==  32. 

32.  5rr  +  33/  +  22  =  217,  40.  Sx  +  Sy  +  z  =  17, 
5^:  —  3y  ==  39,  3:r  +  y  +  3  2  =  15, 
3y  -  22  =  20.  a:  +  dy  +  3z  =  13. 

33.  ^x-^y:=0,  41.    a;  +  2y  — 2;  =  4.6, 
irr-i^^l,  3/  +  22-r?7=10.1, 
^z-y  =  2.  z  +  2x-y^b.7. 

34.  li^r  +  liy^-lO,  42.  a;  +  2y-0.72:  =  21, 
2ix  +  2f  2r  =20,  Sx  +  0.2y  —  2  =  24, 
3iy  +  3f  2  =  30.  0.9a;  +  7y  -  2^  =  27. 


35.    x  +  y  —  z  =  n, 

43.    a;  +  y--=l|2  +  8, 

y  +  z~x=lS, 

3/  +  2=2|y-14, 

z-\-x  —  y  =  n. 

2  +  x  =  3f  a;  -  32. 

36.    a;  +  2/  +  z  =  9, 

44.la;  +  i2/  +  i2  =  36i 

a;  +  2^  +  42=15, 

i^-  +  iy  +  i2  =  27, 

ar+32/  +  92  =  23. 

i^-+iy  +  J-2=i8. 

206  SIMULTANEOUS    LINEAR    EQUATIONS. 

45.  "^-^=2,  49.    --^:^1 

'  +  ?  =  4. 

X       z 

46.  :i:i-L^^2,  50.    ?+l  +  5=:4, 
~  X      y      z 

X     y      z 

9      12_10^^ 

.  X      y        z 

^y   -^ 

x  +  y      b 

y^    ^1^ 

rr  +  2;      6 

zx    __  1^ 

48.    ^+^  =  2,  62.    —^^==20, 

43/  — 3:r 

15, 


y  +  1 

'-'y 

y  +  2 

4 

^  +  1 

^  +  3_ 

1 

rr  +  1 

2 

^x  +  y 

—  9 

z+l 

^y+z. 

—  0 

x-^\ 

^> 

Zz^x 

—  2 

y  +  1 

x-^y  _ 

10 

y-z 

■  •*•'-'> 

X  +  z_ 

-Q 

x  —  y 

■  ^j 

y  +  z 

- 1 

x  +  b 

x  +  S_ 

-2 

y  +  z 

^1 

3/  +  3_ 

-1 

x+  z 

-*-> 

z  +  d_ 

1 

47.  r:_L^  =  io,  51. 


2x-Sz 


y^    ^12. 

a;  +  y      2  42/  — 5z 

53.  (a:  +  2)(2y  +  l)  =  (2a;+7)y, 
(ar- 2)(30  +  1)  =  (a:  +  3)(3z- 1), 
(2/  +  l)(z  +  2)  =  (y  +  3)(z+l). 

54.  (2a;-l)(y+l)  =  2(a:+l)(y-l), 
(a;  +  4)(2  +  l)  =  (a;  +  2)(2  +  2), 
(y-2)(2  +  3)  =  (y-l)(2  +  l). 


SIMULTANEOUS    LINEAR   EQUATIONS.  207 

55.  {x  +  l){by-^)  =  {1x+l){2y-?>), 

G/  +  3)(.  +  2)^(3y-6)(3.-l).       ' 

56.  21^  +  31y  +  422  =  115, 

^(2x  +  y)^^{^x  +  z)  =  2{y  +  z). 

57.  lb{x-2y)=.b(2x-Zz)  =  ?>{y  +  z), 
21a^+31y+41z-135. 

58.  ^x{y  +  z)  =  4:y(z  +  x)=:'^z{x  +  y), 

'-  +  W--9. 

X     y      z 

■       59.    ?>x  +  y  +  z^20,  60.    i^  +  2  +  8y  ==  30, 

?>u  +  x  +  4.y^m,  bu  +  y  +  z-=lO, 

3  w  +  6:r  +  2  =  40,  4:u  +  x  +  z  =  l0, 

5i^  +  8y  +  32;  =  50.  '^u  +  x  +  y=^lO. 

§  45.  The  principle  of  symmetry  is  often  of  use  in  the 
solution  of  symmetrical  equations.  For,  from  one  relation 
which  may  be  found  to  exist  between  two  or  more  of  the 
letters  involved,  other  relations  may  be  derived  by  symme- 
try ;  also,  when  the  value  of  one  of  the  unknown  quantities 
has  been  determined,  the  values  of  the  others  can  be  at 
once  written  down,  etc. 

Examples. 
1-    {x  +  y){x  +  z)  =  a, 

(^  +  y)  (y  +  ^)  =  ^. 

{x+  z){y  +  z)-=c. 

Multiply  the  equations  together  and  extract  the  square 

root. 
.\(x  +  y){y  +  z){z  +  x)-=^{ahc). 


208  SIMULTANEOUS    LINEAR   EQUATIONS. 

Divide  this  equation  by  the  third. 

,\x-{-y^=  ^^ ^  ;  and,  therefore,  by  symmetry, 

a 

b 
Hence,  we  get 

ah  —  hc-\-  ca 

^         2-^/{ahc) 

whence  y  and  z  may  be  derived  by  symmetry. 

2.  x  +  y  +  z^O,  (1) 
ax  +  hy  +  cz  =  0,  (2) 
hex  +  cay  +  ahz  +  (a  —  b){h  —  c)  (c  —  a)  =  0.             (3) 

cx(l)-(2)   gives  (c~a)x  +  (c-b)y  =  0. 

Hence,  y  —  ^ ^,  and  similarly, 

b  —  c 

(a-b)x 

b-c 

Substitute  in  (3)  these  values  of  y  and  z,  and  reduce ; 

then,  x{a--b)(c  —  a)~{a  —  b)  (b  —  c)(c  —  a), 

or,      x  =  b  —  c. 

Hence,  y  =  c  —  a,  z=^  a  —  b. 

3 .  a(yz  —zx  —  xy)  =  b(zx —xy— yz) = c(xy —yz—  zx)  ==  xyz. 
Divide  the  first  and  last  equations  by  axyz ;  then, 

-,  and  hence,  by  symmetry, 


I 


a      X 

y 

z' 

1_1 

1 

1 

b      y 

z 

X 

1_1 

c      z 

1 

X 

1 

y 

SIMULTANEOUS    LINEAE,    EQUATIONS.  209 


Therefore,  -  +  -  =^  —  -,  and,  by  symmetry, 


0       c 

2 

X 

c      a 

2 

y 

a      0 

2 

4.  ax  +  hy  +  cz^l,  '  (1) 
o?x  +  hhj  +  c'z=l,  (2) 
o?x  +  bhj  +  ch=-l.  (3) 

'"       cX{l)~  (2)  gives  a{c  ~  a)x  +  b{c'''b)y^c    -  1.     (4) 
cX{2)-(?>)giYe^a\c-a)x  +  b\c-b)y  =  c—l.    (5) 
^  X  (4)  —  (5)  gives  ab{c—a)x—a\c—a)x—b{c—l)—{c—l), 
or  a(a  —  b)  {a  —  c) X  ^  (c '—  1)  (^  —  1). 

Iherefore,  x  =  -^ f^ \  ', 

a{a—-b){a  —  c) 

whence  y  and  z  may  be  derived  by  symmetry. 

5.  Eliminate  x,  y,  z,  u  (which  are  supposed  all  different) 

from  the  following  equations : 

X  =  by  -\-  cz  +  du, 

y  z=  CZ  -\-  dic-[-  ax, 

z  =  du -\-  ax -}-  by, 

u=-  ax  '\-by  ■\-  cz. 

Subtracting  the  second  equation  from  the  first, 

x~y^=by  —  ax,  or  {1 -\- a) x  =^  (1 -{- b) y , 

which,  by  symmetry, 

=  {l  +  c)z  =  (l  +  d)u. 

These  relations  may  also  be  obtained  by  adding  ax  to 
both  members  of  the  first  equation  by,  to  both  mem- 
bers of  the  second  equation,  etc. 


210  SIMULTANEOUS    LINEAR    EQUATIONS. 

Now  divide  the  first  equation  by  these  equals. 
1      _      h       ,       c     J       d 

And  since =1 ,  we  have 

1  +  a  l  +  (2 

1  a       ,       h       ,       c       .       d 


i 


1+a      1+^      l+c      l+d 


Ex.  67. 

1.  Given  ax-[-hy  =^  c,  and  that  x  =  — -— -, 

a^x  ~\-})hj  ■=  c\  derive  the  value  of  y. 

2.  Given  hx  =  ay,  ,     and  that  x  =  -~ -~\ 

be  —  ad 

dx-{-7nd=cy-{-nd,    derive  the  value  of  y. 

3.  Given  ax  ~{- by  -}-  cz -—  d,     and  that  x  =  — ^ -^ (, 

a  (a—b)  (a—c) 

c^x-\V'y^&z  =  (i^    write  down  the  values  of 
c^x-\-Wy-\-c^z  ^=  d^,    y  and  z. 

4.  There  ia  a  set  of  equations  in  x,  y,  z,  u,  and  lu,  with  cor- 

responding coefficients  (a  to  x^  etc.),  a,  b,  c,  d,  and  e  ; 
one  of  the  equations  is  x)=  by -{-  cz -\-  du  -{-  ew,  write 
down  the  others.  , 

Solve  the  following  equations : 

X    ,  y  y  ,  z       T      X    ,  z 

m      n  n     p  vi     p 

6.  X  -\-  ay  -{-  bz  =  VI,  y  ~{-  az  -{-  bx  =  n,  z-\-  ax~\-by  =p. 

7.  x-\-  ay  -=^1,  y  ~{-bz-=^m,  z  -\-  cu  =  n,  u-\-  dw  ==^p, 

w  -\-  ex-=^r. 


SIMULTANEOUS    LINEAR    EQUATIONS.  211 

8.  Eliminate  x,  y,  z  (supposed  to  be  all  different),  from 

the  following  equations :   x=^hy  -{-  cz,  y  =^  cz-\-  ax^ 

Z'^  ax-\-  by. 

9.  Eliminate  x,  y,  z  from 

X  y  J  z 

I  y+^  z+x  x+y 

10.    Having  given 

1^,  X  —  by  -}-  cz  -\~  du  +  ew, 

y  ^=^  cz  -{-  du -\-  eiv -\-  ax^ 

^  z  =^  du -\-  ew -{- ax -}-  by^ 

u  =  ew  -{-  ax  -\-  by  -}-  cZj 

w  —  ax  -{-  by  -}- cz  +  du, 

show  that  -^  +-A_  +^  +_iL.  .  ^_  :^  1. 

1  +  a^  1  +  6^1  +  ^^1  +  6^^1+6 

§  46.   Resolution  of  Particular  Systems  of  Linear 
Equations. 

Examples. 

1.  x  +  y  +  z  =  a,  (1) 

y  +  z  +  u^-b,  (2) 

z  -\-u+x  =  c,  (3) 

u+x  +  y  =  d.  (4) 

(l)+(2)+(3)+(4)    S(u+x+y+z)  =  a  +  b  +  c  +  d,    (5') 
3(1)  3(:r+y  +  z)=.3a,  (6') 

i[(50-(6')]  ^  =  ^(-2a  +  5  +  .  +  c^). 

The  values  of  x,  y,  and  2;  may  now  be  written  down  by 

symmetry. 
The  following  is  a  variation  of  the  above  method,  appli- 
cable to  a  much  more  general  system. 
Assume  the  auxiliary  equation 

u  +  x  +  y  +  z  =  s.  (5) 


212  SIMULTANEOUS    LINEAR    EQUATIONS. 

Hence,  (1)  becomes  s  —  u~-  a,  (6) 

(2)  becomes  s  —  x  =  h,  (7) 

(3)  becomes  s  —  y=^c,  (8) 

(4)  becomes  s  —  z  =  d.  (9) 
(5)  +  (6)  +  (7)  +  (8)  +  (9),45-.  +  a  +  5  +  ^  +  c^. 
Therefore,                   s  =  l(a  +  h  +  c  +  d). 

s  is  now  a  known  quantity,  and  may  be  treated  as  such, 
in  (6)  giving         u  =  s~~a, 
in  (7)  giving         x  =  s~  b, 
in  (8)  giving         y  =  s-c, 
in  (9)  giving  z=^  s  —  d. 

2.  yz  =  a{y^z),  (1) 

zx^l{z-^x),  (2) 

xy=-c{x~\-y).  (3) 

(X)~ayz,  _  +  -  =  _, 

y      z     a 

^-'   ^-  =  \ 

z~^  X      b' 

{?>)^cxy,  1  +  1=1 

X     y      c 

This  may  now  be  solved  like  Exam.  1,  using  the  recip- 
rocals of  a,  b,  c,  X,  y,  and  2,  instead  of  these  quantities 
themselves. 


(2)  ^  bzx, 


aiu  +  bi(x  +  y  +  z)  ^  Ci, 

(1) 

a^x -]- b^(y -j- z  +  u)  =  c^, 

(2) 

a-sl/ +  h(z  +  u -i- x)  =  c,, 

(3) 

aiZ  +  hi(u  +  x  +  y)  =  Ci. 

(4) 

Assume  the  auxiliary  equation, 

w  +  a;  +  2/  +  2  =  s. 

(5) 

(1)  becomes  biS  —  (bi  —  Wi)  m  =  Ci. 

Therefore,     — ^s-w=— ^J— . 

(6) 

-«1 


SIMULTANEOUS    LINEAR    EQUATIONS.  213 


f      Similarly,  from  (2),  —^  s~x  =  —^ —  (7) 

^  O2  —  ^2  02  —  a^ 

Similarly,  from  (3),  -A_  s~y=  — ^^ •  (8) 

t>3  —  <^3  o^  —  as 

:      Similarly,  from  (4) ,  --^  s~z^  —^^  •  (9) 

(5) +  (6) +  (7) +  (8) +  (9), 

^1  —  ai      Z)2  —  aa      ^3  —  <^3      ^4  —  «4. 

^L+^i__.  (10) 


b^~ai      ^2  —  ^2      ^3  —  ^3      b^  —  a^ 

From  (10)  we  can  at  once  get  the  value  of  5,  which  may 
therefore  be  treated  as  a  known  quantity  in  (6),  giving 

hiS  —  Ci 

h  —  ai 
and  the  values  of  x,  y,  and  z  may  be  obtained  from 
(7),  (8),  and  (9),  or  they  may  be  written  down  by 
symmetry. 

ax  +  b(i/  +  z)  -^-c,  (1) 

ai/  +  b(z  +  u)=d,  (2) 

az  -j-  b  (u~{-  x)  =  e,  (3) 

au+b(x  +  y)=f.  (4) 

Assume                m  +  a;  +  y  +  z  =  s,  (5) 

(l)  +  (2)+(3)  +  (4),  (a  +  2b)s=c  +  d+e+f.  (6) 

Hence,  5  is  a  known  quantity,  and  may  be  treated  as 

such. 
From  (1)  and  (5),  bs  —  bu  -}-  (a  —  b)x  =  c. 

Therefore,               bu  —  (a  —  b)  x  =  bs  ~  c.  (7) 
Similarly,  from  (2)  and  (5), 

bx  —  (a  —  b)y  =^bs  —  d,  (8) 


214 


SIMULTANEOUS    LINEAR   EQUATIONS. 


5. 


From  (3)  and  (5) ,    by-~{a  —  h)z  =  hs-  e.  (9) 

From  (4)  and  (5),    hz-(a~  h)u=  hs -/.  (10) 

&(7)  +  (a-^)(8), 

h^u  —  (a  —  hyy  =  ahs  —  he  —  {a  —  b)  d.  (11) 

S(9)  +  (a-Z>)(10), 

Vy  -  (a  -  Vfu  =  ahs  -he-  (a,  -  h)f.  (12) 

5^(11)+ (a -5)^(12), 

_6[J'^(c-c^)  +  («-6y(e-/)].  (13) 

The  values  of  x^  y,  and  z  may  now  be  written  down  by 
symmetry. 

cc"  +  o^x  +  ay  +  2;  =  0, 
^3  _|_  ^2^  _j_  ^^  _!_  ^  ^  0, 

&  ^  c^x  -\-  cy  -\-  z  ^^  0. 
The  polynome  i^^  ■\- xf  -\- yt -\- z  vanishes  for  t^=^a,  t=^h, 

Therefore,  by  Th.  II.,  page  58,  for  all  values  of  t, 
f'  +  xe  +  yt  +  z  =  {t  —  a)  (t  -h)(t~  c) 

=  f  —  (a  +  b  -{-  c)f  +  (ab  +  he  +  cd)  t  —  ahe. 

Therefore,  by  Th.  III.,  page  67, 
x^  —  ^a  +  b  +  c), 
y  ^=  ab  -\-  be  -{-  ca^ 
2  =  —  ahe. 


6.  x  +  y  +  z  +  u  =  \,  (1) 

ax  +  hy-\-cz  +  du  =  0,  (2) 

a^x  +  b''y  +  ch  +  d''u  =  0,  (3) 

a'x  +  ^V  +  ^'^  +  ^'^  ==  0.  (4) 

Employing  the  method  of  arbitrary  multipliers, 

(4)  +  Z(3)  +  m(2)  +  n(l), 

u  =  n  (5) 


a' 

x+  ¥ 

y+  C 

z^d^ 

+  Id' 

+  lh' 

+  h? 

+  ld' 

-\-ma 

+  mh 

-\-mc 

+  md 

+  n 

+.n 

+  « 

+  n 

SIMULTANEOUS    LINEAR    EQUATIONS.  215 

To  determine  x,  assume 

h^  +  lh'  +  mh  +  n^  0,  (6) 

c^  +  lc^+mc +71=^0,  (7) 

cZ^+  ld''+  md+  n^O.  (8) 

Therefore,       x= -— —  (9) 

a^  +  la^  +  ma  +  n 

But  the  system  (6),  (7),  (8)  has  been  solved  in  Exam.  5, 
from  which  it  is  seen  that 

l  =  —  (h  -\-c-\-  d),  7n  =  hc-\~  cd-{-  db,  n  =  --  bed, 

and  a^  +  a^l  +  am  -j- n  =  (a  — b)(a  —  c)(a  —  d). 

Hence,  using  these  values  in  (9), 

—  bed 
x  = : — 

(a  —  b)  (a  —  c)  (a  —  d) 

The  values  of  y,  z,  and  u  may  now  be  written  down  by 
symmetry. 

^-^-^       +r7^„=l.  (1) 

1,  (2) 


m  —  a      m  —  b      m  —  c 


n —  a      n  • 


b      n  —  c 


X  y  z 


+  _f_=l.  (3) 


p—  a     p  —  b     p  —  0 

Assume  1 ^ 

t  —  a      t  —  b      t~-c 

__  f  +  Bf  +  Ct  +  D  ,4^ 

(t-a)(t-b)(t-c)  ^  ^ 

But  in  virtue  of  equations  (1),  (2),  and  (3),  the  first 
member  of  (4)  vanishes  for  t  =  m,  t-=n,  and  t=p', 
and  hence,  f  +  Bf^  +  Ct-\-  D  vanishes  for  the  same 
values  of  t]  and  therefore,  by  Th.  II.,  page  58, 

f  +  Be  +  Ct  +  n  =  (t  -m){t-  n)  (t  -p). 


216  SIMULTANEOUS    LINEAR    EQUATIONS. 

Therefore,  (4)  becomes 

1  _  _x_ y_ z__  _  (t  ~  m)  (t  —  n)(t  —  p) 

t-a      t-h      t-c~  {t-a)(t  —  b){t~~c)' 

To  obtain  the  value  of  x,  multiply  both  sides  of  this 
equation  by  (t  —  a). 

t-a  —  x-  y^^~^^  -  ^(^-^)  =  (t-m)(t-n)(t-p) 
t—b  t-c  {t-h)(t-c) 

Now  t  may  have  any  value  in  this  equation ; 

let  t  —  a. 

Hence,  x  =  {a-m){a--n){a-p)^       , 
{a  —  b){a  —  c) 

The  values  of  y  and  z  may  now  be  written  down  by 
symmetry. 

8.  ^  +  ^^3/  +  ^^^  +  ^  n\ 

p  q  r 

Ix  +  my  +  nz  =  s^.  .  (2) 

By  §  37, 

x-{-  a y  ~{-b  _z-\~  c lx-{-7ny -{-nz-j- la-j-mb -j- no 

p  q  r  Ip  -\-  "rriq  +  nr 

,o\  s^  +  ^a  +  "mb  -\-nc       r> 

(2)  =  — \ ' ' =  B,  say  ; 

tp  +  mq-\-nr 

therefore,       x=^pR  —  a,  y  =  qR  —  ^,  z^=rJR~c. 

9.  yz  +  zx  +  xy  —  {a  +  b-\-c)  xyz.  (1) 
yz  +  zx  __zx  +  xy  ^xy  +  yz  ^2) 

a  b  c 

(l)^xyz,      \jr\  +  \  =  a+b  +  c.  (3) 

xyz 

1+1    1+1   1+1 

{2)^  xyz,      ^_JL=y—l=l-Jl.  (4) 


SIMULTANEOUS    LINEAR    EQUATIONS. 


217 


10. 


§  37  and  (3), 
(4)  and  (5), 

.  .  -  +  -  —  za 
X      y 

1 


2  2  2 
X  y  z 
a-[-b  +  c 


:2. 


y      z 


21), 


Z        X 


(3) -(6) 

X^  c 


a+b 

X 


X 


^=  a  —  b-{-  c,  -  =  a  +  b- 

y 


-c, 


1 


a-\-  c 


^  I  y-<^ . 


a—c      a—b 


:2. 


(1). 

.  x-\-c 
'  'a  +  b 

-1  =  1 

y  +  ^ 

a  +  c 

,  x  —  a 

-b  +  c 

a  +  c  —  b  —  y 

a 

+  b 

a  +  c 

Similarly,  from  (2), 

x  —  a 

-b  +  c 

a—b  +  c—y 

a 

—  c 

a  —  b 

(3) 

and  (4), 
x  —  a  — 

b  +  c  = 

—+-(a  —  b  +  c 
a  +  c^ 

a  — 

^  +  ^-y)- 

But 

unless  ^~^    = 

a  —  c 

> 

(5) 

(6) 

-a+b  +  c. 

(1) 
(2) 

(3) 
(4) 


If 


a+c      a—b 
this  cannot  be  the  case  except  for 
a  —  b  +  c  —  y=^0, 
in  which  case  x  —  a  —  b  +  c^=^0  also, 
giving     X  =  a  +  b  —  c  and  y  —  a  —  b  +  c. 
a  +  b      a  —  c      ...      7  2        2       2 

I  _  •      --2        .  ^2  _  ^2  _  ^2^ 


a+c      a—b 
b'-c''~-=  0,  or  (b  +  c)(b-c)==:  0. 
Therefore,  b  =  c  or  b  =  —  c. 


(5) 
(6) 


218  SIMULTANEOUS    LINEAR   EQUATIONS. 

But,  if  ^  —  +  c  or  —  c,  (1)  and  (2)  are  one  and  the 
same  equation  ;  hence,  if  (1)  and  (2)  are  indepen- 
dent, (6)  cannot  be  true,  thus  leaving  only  the 
alternative  (5). 

11.  2ax  =  {h  +  c~d){y^z\  (1) 

2hy  =  {c-\^a-h)(z-\-x),  (2) 

{x^y^zy  +  x^-^fA-z'^^{a'^y'-\-&).  (3) 

(1)  and  page  155,  (5), 

_yArZ  ^x-\-y-\-z 


(4) 
(5) 


h-^c  —  a        2a        h-\-c-\-a 

(2)  and  page  155,  (5), 

y        _x+z^x+y+z 
c-{-  a  —  h        2  b        c-{-  a-[-b 

(4),  (5),  and  page  155,  (5), 
.  x-\-y  +  z  _         ^         _         y         _         g 

a-\-h-\-c      h-\-c  —  a      c-\-a  —  h      a-\-h  —  c 
.  x'' 

"  (h  +  c-af 

jx  +  y  +  zy  +  x'  +  y'  +  z' 

{a+h  +  cy+{h+ji-ay+{c-^a-by-{-{a  +  h~-cy 

Keduction  and  (3)    ^il+l+lLt^  +  f+A^l, 
Therefore,  x^  =  {b-]-  c  —  of. 

Ex.  68. 

1.  ax-\-by~  c,  3.    ax  +  by  =  c, 
mx  +  ny  =  d.  mx  +  ny  =  c, 

4.    ?  +  ^-l 

2,  ax  +  by  =  c,  ah 

mx  —  ny  ^=  d.  x  -\-y  ^=  c. 


SIMULTANEOUS    LINEAR    EQUATIONS.  219 


5. 


:r   ,   ?/       1  ^^      X      a 


a 


b  y      b 


X  _.y 1  X  -\-71l G 

b      a  y  -\-n      d 


6.  ^  +  f  =  l, 
a      b 

b      a 

7.  ax  -\~bc=^by  -\-  ac, 
x  +  y  =  c. 


Q     a      b 
X      y 

b  ,  a 

-4- -  =  71. 

X     y 


(a-{-c)x —(a  —  c)y  =  2ab , 
(a-{-b)y — (a  —  b)x  =  2ac. 


10. 


a 


12. 

x+y_a+b+c 
y+l      a-b+c 

y~l      a—b~c 

x  +  1      a-\-b-^c 

13. 

x  —  a-\-c  __b 
y  —  a  +  b      c 

y+b      c+a 

x+c      b+a 

14. 

x  +  c   1  y  +  b  _c, 
a-\-b      a-^c 

x-b      y-c__c, 

a~  c      a  —  b 

15. 

^   1   y  __ 

m~a      m-~b 

"  +  y^i. 

y~c      b 

X  —  y  =  a~b. 

16.  x  +  y  +  z^O, 

(b  +  c)x  +  {a  +  c)y  +  {a  +  b)z^O, 
box  +  acy  +  abz  —  1. 

17.  x-\-y-\rZ  =  l, 

ax-^-by  +  cz  =  m, 

^     \     y     \     ^    ^l 
I  —  a      l  —  b      l~  c 


220  SIMULTANEOUS    LINEAR   EQUATIONS. 


18. 


19. 


X  —  a      y  ~h      z  —  c 

l(x  ~  a)  -{-  mi^  —  b)  -\-  n{z  —  c)  ^=  \. 

X  ~a      y  ~h      z  —  c 

Ix  +  my  ■\-nz--=^\. 


20.  a{x  —  a)=^h{y  —  h)=^c{z~c)^ 
ax -\~  by -\- cz  =  w}. 

21.  x-{-y-{-z^=^a-\-b-\-c, 

bx -\-  cy -]- az  =  a^  -{- b"^  -{-  c^, 
ex -^  ay  -{- bx  =  a}  -{- b'^  +  c^. 

22.  x  +  y-^z  =  a  +  b  +  c, 

ax '\- by -{- cz  =^  ab -\- be  +  ca, 

(b  —  c)  X  -}-  (c  —  a)  y  -\-  (a  —  b)  z  =^  0. 

23.    X -{- y -\- z  =  on,  24.    ax~\-by  -{-  cz  =  r, 

X  :  y  :  z  =  a  :  b  :  c.  nnx  —  ny,  qy  =^pz. 

25.  xy  -{•  yz -{- zx  =  0,  ayz  +  bzx  +  cxy  —  0, 

bcyz  +  acxz  +  a^rry  -\-(a  —  b)(b  —  c)  (c  —  a)  xyz  ==  0. 

26.  (a  +  5)a;  +  (^  +  c)y  +  (c  + a)z  =  ab  +  bc-{-  ca, 
(a  +  c)x-{-(a  +  b)y-\-(b-}~c)z  =  ab  +  ac-{-bc, 
(b  +  c)x  +  (a  +  c)y  +  (a  +Z>)2  -:  a^  +  b'  +  c^ 

27.    mx  +  ny  +  pz  +  qu  =  r, 
^  _  y  _  ^  _  ^ 


SIMULTANEOUS    LINEAR    EQUATIONS.  221 


28.   ^(3/  +  ^) ^ y (^ + ^) ^  ^(^  +  3/), 
a  b  c 

--\ \--=^a  +  b  +  c. 

X     y      z 

29.    {a  —  h)  {x  +  c)  —  ay+hz  =  (c~a){y'{-h)--cz-\-ax  —  0, 

x  +  y  +  z^2(a  +  h  +  c). 

35.    l  +  l-l^l 

y      z      X      a 

z      X      y      b' 
1      1_1__2 

X      y      z    ~  c 


30. 

ax-\-by  —  1, 

by  -\-  cz  =  1, 

cz  -\-  ax=  1. 

31. 

ly  +mx  =  n, 

nx  -\~  Iz    =  rti, 

7)iz-\-  ny  =^  I. 

32. 

X  -\~  y  =  a, 

y  +  ^=^, 

X  -{-  Z  '=  c. 

33. 

,                  Tnn 

y+z    x=  ^ 

In 

z-\-  X  —  y  =^  — ' 

Im 
x  +  y-z^-. 

34. 

W-2a, 

y    2 

Z         X 

X      y 

36.  (a+b)x+(a~b)z  =  2bc, 
{b  +c)  y  +{b—c)  X = 2ac, 
(c+a)z  +(c—a)y  =  2ab. 


V     z 

37.    x+Y =  a, 

o      c 

y-\ =  ^, 

^      c      a 

,  X      y 

Z^ T  =  C, 

a      b    ' 


38.    ^_  +  -X_  =  5-a, 


b  -{-  c      c-\-a 
X       .       z 


b  -{-  c      a  +  b 


^=  a  —  Cj 


-JL-^.^-  =  c-l. 
c -\-  a      a-^  b 


222  SIMULTANEOUS    LINEAR   EQUATIONS. 

39.    X  -\-  y  —  z=^  a,  40.    u-\-  v  —  x  ^=  a^ 

y  +  z  —v  =  h,  V  -\-  x~y  =^h, 

z  -{-  V  —  x-=  c,  X  -\-  y  —  z  ^=  c, 

V  -{-  X  —  y  ^=  d.  y  -\-  X  —  u  =■  d, 

z  -j-u—v  =  e. 


Ex.  59. 

Resolve : 

1.    (a  +  h)x  +  (a~b)y  =  2(0" +  b''), 
(a  -  h)x  +  (a  +  Z>)y  -=  2(a'  -  b'). 

2.  x  +  y=^a,  4.    ^a~b)x+(a+b)y  =  a+b, 

X^  —  y"^  z=b. 

_j^^ y   -    1 

3.  2x  —  Zy=m,  a  +  5      a~-b      a-\-b 
2x^  —  Sy'^  =  71^  +  rry. 

e/        z>\i  a  +  ^  +  1 

a~  0 
£ g  -f~  ^  ~  <^ 


7. 


x~y      b  —  c  X  —  y  —\ 

^+A  =  U+A.  ^  +  y  + 1  _  ^^ 

a  +  &      a  +  c  a;  +  3/—  1 

y  —  a     a  +  b  x~y  +  l      a  —  1 

X 


y      d?  -\-b^  X  —  y  —  \      \  —  b 


SIMULTANEOUS    LINEAR    EQUATIONS.  223 

11.  ^  -  y  +  1  -  a^  IS-    {ct--h){x+c)-ay+hz=0, 
'    ^  +  y—^         '  {c~a){y+h)-cz+ax=0, 

^  +  y  +  ^^l^  X  +  y  +  z  =  2{a  +  h  +  c). 

x~y  —  1 

19.    -^  +  ^=a  +  h, 

12.  —--  +  .^J—=a  +  h, 

a  +  b      a-h  y      ^       ^     ^.h  \  c, 

X  ,  y      o  ^+^      ^-^ 

-  +  ^==2a. 

a      0  z       .       X  . 

— — +  - =c  +  a. 

a-f-  0      b  —  c 

13.  {a-\-c)x-{-(a—c)y=^2ah^ 

(a+b)y—la--b)x=2ac.    20.    — — | — '^ ^-=0, 

b-\-c     c  —  a     a  —  b 

14.  o}  +  ax  +  y  =  0,  ^- V- f--A_:^0, 

^2_|_^^_|_^_0.  ^-^     ^-^     ^+^ 

15.  y-\-z~~x=-a^  6  +  c     c— a     a+^ 


2; +07  — 3/  =  ^, 


z 


21.    re  +  _^_^4__i_^l, 


16.    7:.+  lly+.  =  a,  |  +  _X_+_^^1, 


7y+llz  +x  =  b, 
7z  -{-llx +  y~c. 


c      c  —  1      (?  — 2 


17.    ^  +  ^-?^2a5,  22.    -^  =  a 

x  '  y      z  x  +  y        ' 

c      a      b       ^  zx 

z      X     y  z+x 


a      b      c      0       c      a       c      a      b      a      b       c 


224  SIMULTANEOUS    LINEAR   EQUATIONS. 


24. 

X  __y z u 

abed 

31. 

X  +ly  =.a, 
y  +  mz=  b, 

£  .  ^  .  ^  .  !f 

_1 

z  -\-  nu  =  c, 

m      n     p      q 

r 

u  +  pv  =  d, 

25. 

ax=^by  ■='cz=^ 
y^  —  '^  —  x  —  u. 

du, 

32. 

V  -\-  qx  =  e. 

26. 

y  +  z=-au, 
x-\-z  =  bu, 
x-\-y  ^=  cu, 
1  —  x      a 
\-y      b 

y  +  z+u=b, 
z  +  u  +  v  =  c, 

V  -{-  X  -\- y  =^  e. 

27. 

x  +  y  =  m, 
y  -\-  z  =^n, 
z  -}~u=^  a, 
u  —  x  =  b. 

33. 

X  —  y  -\-  z  ^=  a, 
y  —  z+u  =  b, 
z  —u-{-v  =  c, 
u  —  v-\-x  =  d, 

28. 

llx+9y  +  z- 

-u  =  a, 

V  —  a;  +  3/  =  6. 

\ly-\-9z  +  u—x  =  b, 

\lz+9u  +  x  —  y=^c,      34.    x  +  y  +  z—u  =  a, 

llu+9x  +  y~z  =  d,  yj^^j^u-v-=^b, 

29.    x+ay+d'z-\-d'u+a^=0,  z  +  u  +  v-x  =  e, 

x+by+bh+Pu+b*  =0,  u  +  v  +  x  — y=^d, 

x+cy+ch+c^u+c*  --0,  v+x  +  y  —  z=e. 
x+dy+^z+(^u+d'==  0. 

35.    x-j-y  +  z  —  u  —  v—a, 

^        '  y  +  z  +  u-v-x  =  b, 
y  +  z=b, 

z  +  u^c,  ,+u  +  v-x~y^e, 

u+v  =  d,  u  +  v  +  x-y-z  =  d, 

V  +  x  =  e.  v  +  x  +  y  —  z—u  —  e. 


SIMULTANEOUS    LINEAR    EQUATIONS.  225 

36.  2x  —  y  —  z  +  2u  —  V  =  Sa, 
2y  —  z  —  u  -{-  2v  —  X  =^  Sb, 
2z  —  u  --  V  -{-  2x  —  y  =  3c, 
2u  —  V  —  X  ~\'  2y  —  z  =  3c?, 
2v  —  X  —  y  '\-  2z  —  u  =  Se. 

37.  V  ~  2x  -{-  Su  —  2y  -{-  z  =  a, 
x-2y  +  3v  —  2z  +  u^b, 
y  —  2z  +  'dx  ~  2u  +  V  —  c, 
z-2u  +  ?>y~2v  +  x  =  d, 
u  —  2v  +  Sz  —  2x  -\-  y  =  e. 


n 
—  c  —  n 


OHAPTEE  VII. 

PuEE  Quadratics. 

§  47.    (A)    If  an  equation  reduces  to  the  form 
(j}ix  4-  ny  =  c^, 
then  (mx  +  n)^  —  c^  —  0. 

Hence,  (mxi  ~\-n)  —  c=^0,  and  therefore  Xi  = 

or  (mx2  -}-  n)  ~\-  c  =  0,  and  therefore  X2  = 

IIV 

{B)    If  an  equation  reduces  to  the  form 

fmx  +  ^Y_  o? 
\px  +  q)~  h''' 

tlipn  X  -^^-^^  _-qa-nh 

tilt;  11  jji  —       -  ,     jj2 ^ • 

mb—pa  inh-\-pa 

(See  Exams.  4  and  5  below.) 

Examples. 

^     x  +  ?>{a  —  h)_a(?>x  +  ^a-lh) 
x  —  3(a-h)      b(Sx~7a  +  9b) 

Apply,    if  !^^-£,  therefore  !?L+^:^^£±f. 
n      q  7n  —  np  —  q 

Hence  ^        3^  3r^(a  +  ^)  +  9a^  -  14a&  +  9^^ 

'      S(a-b)  Sx(a~-b)  +  9(a'-b') 

Dividing  the  denominators  by  S(a  —  b), 

'  x[x  +  S{a  +  b)]  =  Sx(a  +  b)  +  da"  -  Uab  +  9b\ 

Therefore,  x'  =  9a''-Uab  +  9 b\ 


PURE    QUADRATICS. 


2. 


2a  +  4:by_bx—da  +  3b 


Apply,  if  — =-2,  therefore  =  I — ^, 

n       q  n  p 

and  factor  the  numerator 

{x  +  ^a-  2by  -{x  —  2a  +  Uy- 

.   l'2{x+a  +  b){a-b)_     I2{a~b) 
{x  +  ^a-2by  bx  +  ?>a-9b 

.      x  +  a  +  b    _  x  +  4:a  —  2b  _     S(a~b) 
"  x  +  4:a  —  2b      6x  +  Sa-9b      4:X~a—7b' 

by  taking  difference  of  numerators  and  difference  of 
denominators. 

To  the  first  and  third  of  these  fractions  apply,  if 

^=■2,  therefore  ^^=^. 
n      q  n  —  m      q~p 

x-\-a-\-b 3(a- 


'  3(a-^)       ^x  —  ^a-^b 
:A[x'-{a  +  by']'-==9{a-b)\ 
.'.x'^\[^{a  +  by  +  9{a-hyi 


{x  +  2y^a 
x^  —  2x      b 


{x+2y 


= ^ (1) 

m{x-{'2y -\-n{x'^ —  2x)      ma+nb 

But  {B)  can  be  applied  if  m  and  n  are  so  determined 
that  m{x-\-2y  -\-n (x^  —  2:^)  is  a  square. 

This  requires  that  4  m  (m  +  n)  =  (2  m  —  nf. 

.'.  4 m^  +  4 mn  =  4m^  —  4mn  +  n\ 

.'.  8m  =  n. 


228  PURE    QUADRATICS. 

Assume  m  =  1,  then  n  =  8,  and  (1)  becomes,  on  substi- 
tution and  reduction, 
{x  +  2y  _      a      _   2 
(3;r-2)'~^r+86"'''  '^^• 

.„_2{l+r)  2(r-l) 


(^+ir 


a 


"  {x'  +  l){x^-2x+l)      b 

Yov  x^  -\-l  write  ^r^;. 

.     {xz  +  2xy  ^  a^       .    {z  +  2y  ^  a 
xz{xz~2x)      b  z(z  —  2)      b 

This  equation  was  solved  in  Exam.  3,  hence  z  may  be 
treated  as  known. 

But^^ii--         .  x'  +  2x  +  l_z  +  2 
"x^-2x  +  l      z~2 

2  +  2 


-,  a  form  solved  in  {E). 


Ex.  60. 

1.  {x  +  a  +  b){x~-a  +  b)  +  {x  +  a  —  b){x  —  a  —  b)^0. 

2.  {a+bx){b  —  ax)  +  {b  +  cx){c—bx)-\-{c+ax){a—cx)  =  0. 

3.  (a  +  5a;')  {ax  —  b)  +  {b+  ex)  (bx  —  c)  +  (c  +  ax)  (ex  —  a) 

=  i(a^  +  b'  +  e'), 

4.  (a  +  x)(b-x)  +  {l  +  ax)(l-bx)^(a  +  b)(l  +  x'). 

5.  (a  +  x)(b  +  x)(e  ~  x)  +  (a  +  x)  (b  ~x){e  +  x) 

+  {a  —  x){b  +  x)  {c  +  x)  +  {a  —  x)  (b  —  x)  (c  +  x) 
+  (a  —  x)(b  +  x)  (e  —  x)  +  (a  +  x)  (b  —  x)(e  —  x) 
—  babe. 


PURE    QUADRATICS.  229 


6.  {a  +  x)  (b  +  x)  {c  +  x)  +  {a  +  x)(b  +  x)  (c  —  x) 

-{- (a -\-  x)  (b  ~  x)  (c  +  ^)  +  (^  —  ^)  (^  +  ^)  (<^  +  ^) 
-]-  (a -{- x)  (b  —  x)  (c  —  x)-\-(a  —  x)  (b  +  x)  (c  —  x) 
+  (a  —  a;)  (b  —  x)  (c  -{-  x)  -\-  (a  ~  x)  {b  —  x)(c  —  x) 
=  Sx\ 

7.  {a  +  bb  +  x){ba+b  +  x)=-?>{a  +  b+x)\ 

8.  {a  +  l1  b  +  x)(\l a  +  b  +  x)  =  ^(^ci  +  b  +  x)\ 

9.  {9a—1b  +  ^x){^b-1a  +  ^x)-=-(2>a  +  Zh  +  x)\ 

1 0  Q?^       _[_       cd       _  Q     -  _      g  —  a:  _  b  —  x 
o?—h^x^     c^  —  d'^x^        '  1  —  ax      1  —  bx 

11  ^^  —  Q^  I  x-\-  a  _cy  ^n.     x-\-a+'2b  _b  —  2a-\-2x 
'    .r+1      x-l^"^'  '    x+a—2b~ b  +  2a—2x 

-2     a  +  x  _x  +  b  a+^b+x _?>b  —  a+x 

a  —  x      x  —  b  a  —  4:b+x      ?)b-\-a  —  x 

^„     ax -\-b  _  ex  ~\-  d  -J.     x-\-ba-{-b x—a-]-b 


a-\-bx      c-\-dx  x~Sa^b      a—x^3b 

14      <^  —  ^  _  1  —  bx  a  —  7 b  +  x a-j-bb-j-x 

1  —  ax       b  —  x  la  —  b—x      ba-\-b-[-x 


20. 


21. 


22. 


23. 


24. 


?)a—'b^x bb  —  ^a-\-  X 


a  — 

-3b  +  x 

ba  —  db  +  x 

3  a 

-2b  +  Sx 

■_    x  —  a  +  2b 

a 

--2b  +  x 

3x~Sa  +  2b 

3a 

~2b-\~3x 

_  x—7a  +  8b 

a 

-2b  +  x 

3x--  ba  +  4:b 

5a 

-6b  +  x_ 

__3a~-5b  +  Sx 

a  -{-  X 

a  +  b-}-x 

a 

+  b~x 

_S(a-b  +  x) 

3a~  b  —  Sx       a  —  bb -{-X 


230  PUHE    QUADRATICS. 


25         "^  a  +  h  —  X     _  3  (g  —  /;  -f  x) 
ba  +  ?>h  —  ?>x~  a~llb  +  x' 

2g        ba-h  +  x    ^  2{2a~h  +  x) 
'  ■'2(ia4-2b-x)        a+llb~x  ' 

„,-     7 a  —  h  -\-  X  __  a(a  -j-  bb  -\~  x) 
1b-a  +  x~b{ba  +  b+x) 

28     ^  +  <^ -—  ^  _.  a{x-{-  a-\-bb) 
X  ~  a  +  b      b  {x -{- b  a -{- b) 


29. 


30. 


31. 


32. 


rba-U  +  xV_^  1a-^b  +  ?>x 
\bb~?>a  +  x)'    lb--'da  +  ?>x 

'a  +  bb  +  x\_a  +  11b  +  x 


ba-\-b  -\-  xj      lla-\-b-\-x 
b  +  x\__l1a  +  b-x 


/7a 


a  +  xj      VJb-{-a  —  x 


l1a  +  b-x^o?(a  +  l1b  +  x) 
a+llb-x      U'{Vla  +  b  +  x) 


33  {x  -\rl  a  +  b){x  —  a-\-b)       __      x  —  ba~\-b 
{bx  +  ?>a-llb){x-a+llb)^  bx+la-b2b 

34  (l  +  3^+5.2:^)(^^  +  3:r+5)_9 
{l  +  2x  +  ^x'){x'  +  2x-[-?>)      4* 

35.    ~ I + 1 = ^^ 


:?;"^- 11a; +  28      x'-llx+IQ     x^--l4:x  +  ^0 


36.    ^ + ^ = ± 

x'-^x  +  b     x'  -  Ux  +  45      x'-  llo;  +  10 

37.  x\b  -  a')  +  a\x  -  b')  +  Z>X«  -  ^')  +  a^^(«^^  -  1) 

=  (a-x')(b'-a'). 


QUADRATICS.  231 


> 


Quadratic   Equations   and   Equations  that  can  be 
resolved  as  quadratics. 

§  48.    ((7)    If  an  equation  appears  under  the  form 

(a  —  x)  (x~h)  =z  c,  (1) 

then  ^1  =  |- (a  +  5  +  r),  x.2=^^(a~{-  b  —  r), 

in  which  ?^^  =  (a  —  by  —  4  c. 
From  the  identity 

(a  —  x)  -{-  (x  —  b)  =  a  —  b 
W9get      (a-xy  +  2(a-x)(x-b)  +  (x-by--=(a~by,    (2) 
(2)-4(l)  {a~xy  -2(a-x)(x-b)  +  (x  -by 
=  (a  —  by  —  4tc  =  r^  say. 
Then,   [(a-  x)  -  (x -  b)J  -r'  ==  0] 
hence,       [(a—Xi)—(xi—b)]-]-r  =  0,  and  .\Xi  =  |-(a-t-Z>+r)  ; 
or,  [(a-~X2)~-(x2—b)]—r  =  0,  and  .'..X2='  ^(a+6— r). 

Examples. 

1 .    x'  +  {ab  +  ly  -  {ciS+  b')  (:i;*^  +  1)  +  2  (a'  -b')x+l. 
.■.x'  +  a'b'  =  (a'  +  b')  x'  +  2 (a'  -b')x  +  (a-  by. 
/.x'+2abx'+a'b'  =  (a+byx'  +  2(a:'-b')x  +  (a—by. 
.-.  x^  +  ab  =  ±[(a  +  b)x  +  (a  —  b)], 

or  x'^  ^  (a  -}-  b) X  -{-  ab  =  dz  (a  —  b). 
.'.y  ^{a+b)x  +  \{a  +  by  =  \{a-by^{a-  b). 
.'.x-^^{a-^b)  =  \^[{a-by±^{a-b)l 


2. 


ax-^b mx  —  n 

bx-\-  a      nx  —  m 

Add  and  subtract  numerators  and  denominators, 

(g  +  Z^)  (.r  +  1)  _,  (m  +  n)  {x  —  1) 
(a  —  ^)  (:?;  —  1)      (m  —  n)  (:^  +  1) 


232  QUADEATICS. 


f^±l\'=  {a-b)(^^  +  n)  ^ ,._  3^^, 


"\x~lj      (a 


-j-b)(m  —  n) 


t  +  1  _s-l 


S—1  5+  1 

3.  (a~xy  +  (h-xy  =  c. 
In  tlie  identity 

(u  +  vy=::u'  +  v'  +  4:(u  +  vfuv  -  2t^V 

let  u  =  a  —  X,  V  =  X  —  b.  , 

.'.u-\-v—a  —  b  and  u^-{-v^^c. 

/.(a-by  =  c+4(a~by(a~-x)(x-b)-2(a-xy(x--by. 

Write  z  for  (a  —  x)(x~b). 

.'.  z'  -2(a-  byz  +  (a-  by  =  ^[c  +  (a-  by]  =  f,  say. 

.•.[z-(a-byy=:f. 

.'.hj(JBy  z,  =  (a~by-t-  z,  =  (a-by  +  t. 

.'.  z  is  known. 

But  (a  —  x)(x~b)  =  z. 

.•.by((7),  x,^^(a  +  b  +  r);  :r^  -  |(a  + 5  -  r),  (1) 
in  which  r^  =  (a~by  —  4:Z 

=  (a~by-4:[(a-by-t]--=At-S(a-by\  .ox 

0T(a-by-A[(a-by+t]  =  -4:t-^(a~by)  ^  ^ 

SindLf  =  ^[c  +  (a-byi  (3) 

Hence,  x  is  expressed  in  terms  of  a,  b,  and  r  ; 
r  is  expressed  in  terms  of  a,  b,  and  t ; 
t  is  expressed  in  terms  of  a,  b,  and  c ; 
and  the  expressions  for  r  and  t  are  cases  of  (A). 

4.  (a  -:t')  (b  +  ^^  +  (a  -  a;/ (5  +  x)=ab  (a'  +  ^^'). 
Let  a  —  x^=n~~  z  and  5  +  ^  —  '^  +  2;. 

.■.w  =  |(a  +  J).  (1) 


QUADRATICS.  233 


The  equation  reduces  to 

(n^  -  z')  [(n  +  zf  +  (n-  zf]  =  ah  (a'  +  ^'). 
.-.  (n'  -  z')  (2n'  +  6nz')  =  ah  (a'  +  h'). 
.-.  (n'  -  z')  (V  +  Sz')  =  ah  (a'  -  ah  +  h'). 
'£■  may  now  be  found  by  (C),  and  from  (1) 

x=^\{a—h)-\-z, 

Zz^^\{a-lf  or  \{\^ah-a^-h''). 
:,x  =  ^,  ova-h,  ov  ^{a-h)+\^{?>0ah-'S>a^—2>h^), 

5.    rr*-4  =  ^-±^;  .-.x^-^x'-bx'-U^O. 
x^-2 

Find  the  rational  linear  factors  of  the  left-hand  member 

by  the  method  of  §  27,  page  116. 

.\{x-2){x  +  2){x'  +  2x^  +  ^)  =  0. 

:,x-2  =  0,  or  rr +  2^:0,  or  :r*  + 2.^^  +  3  =  0. 

The  last  of  these  equations  may  be  solved  as  a  quad- 
ratic, giving 
:i;'-:-lzb2V-2. 

.-.rr  =±l±V-2. 

:.x,  =  2-  x,  =  -2)     x,=  l  +  -yJ-2-    x,=  l~-yJ-2) 
^5  =  -l+V-2;  x,  =  -l--^~2. 

Note.  In  solving  numerical  equations  of  the  higher  orders,  the 
rational  linear  factors  should  always  he  found  and  separated^  as  dis- 
junctive equations,  before  other  methods  of  reduction  are  applied. 
Such  separation  may  always  be  effected  by  the  methods  of  §§  26-29, 
and,  unless  it  is  done,  the  application  of  the  higher  methods  may. 
actually  fail.     Thus,  if  it  be  attempted  to  solve  as  a  cubic  the  equa- 

^^^^  0,-3 -9a; -10  =  0,         ^ 

the  result  is  x  =  {5 -\- y/-  2)h  +  (5  -  V"  2)3, 

which  can  be  reduced  only  by  trial.     The  left-hand  member  can, 

however,  be  easily  factored  by  the  method  of  §  27,  and  the  equation 

^^^^^^^  ^0  (x  +  2){x^-2x~d)^  0, 

which  gives  .t  =  2  or  1  ±  y/6. 


234  QUADRATICS. 


6.    (x-2y~x'  +  2'  =  0. 

Factor  (see  Exam.  20,  page  113),  rejecting  constant  fac- 
tors, 
.-.  x(x  —  2) (x^  —  2x  +  47  =  0. 
...x^O,  or  x  —  2  =  0,OY  x''-2x+4:^0. 
The  last  equation  gives  r^  ==  1  ±  V~  ^• 

Ex.  61. 

Solve  the  following  equations : 

1.  (x  +  a  +  bf  =  x'  +  a'  +  b\ 

2.  (x-i-a  +  hf  =  x'  +  a'  +  b\ 

3.  (a  —  h)x'  +  (b-x)a'  +  (x-a)b^  =  0. 

4.  (a  —  b) x^  -j-(x  —  b)  c^  ■\-(x-\-d)  b'^  —  2abx. 

5.  {x-~af  +  {a-by  +  (b  —  xy^O. 

6.  {x  -  aj  +  (a  -  by  +  {b  -  r?;)^  =  0. 

7.  (a^  -  Z^) :r*  +  {x^  -  a)  b'  +  (P  -x)a'  =  abx (a^b'^x'  - 1). 

8.  {x  ~  a){x  —  b)  {a  —  b)-\r{x  —  b)  (x  —  c)(b  —  c) 

+  (x  —  c)(x  ~  a)  (c  ~  a)  =  0. 

9.  ?l_i=:0.  11. ±  =  0. 

x—1  x^—1 

10. ^  =  0.  12. -  =  0. 

x^  —1  x*—l 

13.  x'  +  5a^~16x''  +  20x~16^0.     (See  §  21.) 

14.  x*-Sx^  +  5x''-\-6x+4:  =  0. 

15.  (x-ay  +  x'  +  a'  =  0.       17.    rr  (r?;  -  2)^ (re  +  2)  -  2. 

16.  2a^  =  {x~6y.  18.    (4rr^-17):r+12  =  0; 
19.  :r*+  (a^>  +  l)'^  -  (a'  +  b')(x'  +  1)  +  2(a^-  ^^^)  a:  +  1. 


QUADRATICS.  235 


20.  x''  {x  -  169)2  -f- 1 7  .r  =  x''  -  3540. 

21.  Qx{x''  +  Vf  +  {2x'+by=lb0x+l. 

22.  2;r(:r-l)2  +  2-=(.^'+l)^     24.    5^;^=  12.x^^  + 1. 

23.  .X'*=12:i;  +  5.  25.    {x  + ^f  ='^{2x ~-l)\ 

26.    ^  +  _^+_^  +  _^  +  _L.+_^___0. 
^'      .r  —  1      X  —  2      X  -  3      .^'  —  4      x  —  b 

27  (^+iy         _m  3Q     {x'+l){x'+V)_on 

{x'  +  l){x-iy'       n  '    (x'-l){x'-l)       n 

28.     (^+1)'  ^^  31.    (^'+l)C-^^+l)_m 

2^     vV  +  l)(rr'+l)_m  32  (^'  -  1)'^  _m 

{x  +  l){x'+l)      n  '    x(x'  +  l)(x-iy      n 


33. 


34. 


35. 


36. 


37. 


38 


x{x-\-\y'       n(ji  —  m) 

{x"  +  1) (x  -  Yf  ~  2m{27n  -  n) 

{x^+iy  _     4m^ 
X  {x^  —  1)^      m^  —  n^ 

{x-l){x''  +  Vf  _2{m-n)\ 
{a^  —  1)  {x  +  1)^  mn 

x^—1         _     2m 
(a:+l)(:r^-l)~2m-7i' 

(a;^  —  1)  (a;  +  1)^  _  m  +  w 
(a;^  +  1)  (r?;  —  17  ~  m  —  92 

(x+  1)  (a;'  +  1)  _  m  +  71 


(a;- 

-l)ix*- 

-1) 

r>i~n 

39.    r'-f -^. 

41.    ;.^-f-^ 

6>:r  —  a 

40.    .r'_«^-^ 
bx~a 

42.    .^...-«^'+^^  +  ^ 
a  +  So;  +  cx^ 

236  QUADRATICS. 


43.  x'  =  (x-iy(x'  +  l). 

44.  aV  =  (a-xy(a'--x''). 

45.  x''  =  (x-ay(x'-l), 

46.  7)i(x  -\-  TTb  —  n)  (x  —  7n  -{•  7  ny 

=  n(x  —  m -\- n) (x  -}-  7 m  —  rif. 

47.  7n^  (x-\-7n-\- 17 n)  (x —  7n  — 5 ny 

=  n^(a7+  17  m  +  n)(x  —  5m  -\-  ny. 

48.  7n?  (x -{- on -}- 17 n)  (x  —  on -\- 7 ny 

=  n^(x-{-17m  +  n)(x-}~7m  —  ny, 

49.    _4-_=r_.  61. L_^ —  =c. 

X     a      n  x^~ax  +  a^ 

x"^  —  ax-\-o?       '  {x-\-  ay 

a—x.x—h      m 


53. 

x~  b      a  —  X      n 

g^     2a?  +  a{a-  x)  +  {a  +  xy  _c  +  1 
2a^  +  a{a  +  x)  +  {a  —  xy      c—1 

55.  x'^-{-(a  —  xy  =  c. 

56.  x^  +  {x-^y^S2. 

57.  {a~xy+{x-hy=-e, 

58.  :?;H(a  — ^)^=-a^;   ^^+(6 -^7=:=  1056. 

59.  (a  -  rr)^ (a;  -  hy  +  {a-  xy  (x  -  by  -  a'h'  (a  -  h). 

60.  {x-a+hy-{x-ay  +  {x-hy-:(^  +  a''-{a-hy--b^ 

=  {a-h)c\ 

{a-xy  +  {x-hy_a'  +  h' 
{a-xy  +  {x-hy      a^  +  h' 

^^     (a-xy  +  {x-hy_a'  +  b' 


(a  -  xy  +  {x--  by 


QUADRATICS. 


237 


63. 

{a-xy  +  {x-hy      a'-b' 

64. 

(a-xY      (b~xf_a'      P 
b  —  X          a  —  X        b       a 

65. 

a  —  X     .     X  —  b    a      b 

{x-by      (a-xy      b'      a' 

66. 

(a-xy  +  (x-by_  a'  +  b' 
(a  +  b-2xy          (a  +  by 

67. 

(a-xy  +  (x-by_  a^-b' 
(a  +  b-2xy          (a  +  by 

68. 

(a-xy  +  ix-by    ^      ^ 

69. 

(a-xy~(x-by__       (a-b)c 

(a  —  x)  —  (x  —  b)       (a  —  x)(x  —  b) 

70. 

(^_^)5  +  (             5)5          ^ 

71 

{a-xf^ix-Vf                c 

(a  -  xy  +  {x-  by '    {a-x){x-  b) 
72.    (l+xJ^{x'-Zy. 

73.        ^'+-^ 


^xix'+l)      b 

^4     {x  +  \y{x^  +  V)  _ 
{x-iy{x^-x+i) 

{x'-x  +  iy~~b 

'  (x+iy{x'  +  i)    b 

^^     (x'  +  iy  _a 

x(x  +  iy    b 


78  (^+^y  ^^ 

x(x'  +  l)      b 


79     •'^(^+1)'  — ^ 

(x  -  ly     b 


80. 


81. 


'x''+x+rTx'-x+T\_  o. 

L(.T+i)'^jLcr-i)^J  'b 

x^  —  x^ -\-\  _a 

{x^-iy  ~~  b 


82     ^(^'  +  1)  — ^ 


238                            SIMULTANEOUS  QUADRATICS. 

33     (x+l)(x'  +  l)_a  gg     ix±iy^a 

(a;-l)(r^-l)      /;  '     x' + 1        h 

\x-\){x'-\-\)      b  '      x'^+l        h 


§  49.    Quadratic  Equations  Involving  Two  or  More 
Variables. 

1.  {x  +  y){x'  +  f)  =  a,  (1) 

x^y  +  xy"^  =  c,  (2) 

(l)  +  2(2),   .•.(a;  +  y)'  =  a  +  2c. 

■.x  +  y  =  -^{a  +  2c).  (3) 

(Any  one  of  the  three  cube  roots.) 

(l)-f-(2),         a^  +  y' __  ft  .    .  /'^^ly Y_ f^^Il^?^. 


By  (3),  :,-y  ^y(«-2.X 

^^^  ^       ^(«  +  2c) 


:  +  y/      a  +  2c 
l£)         . 

Also,  x  +  y  =2li2±M. 

^^       ^(a  +  2c) 

.  V(a  +  2c)  +  V(«-2c) 

2^(a  +  2c) 

■  ._V(«  +  2g)-V(a-2c) 

^  2^(a  +  2c) 

Not  any  one  of  the  six  sixth-roots  of  a+2c  may  be  used 
indifferently  in  the  denominator,  but  only  any  cube-root 
of  whichever  square  root  of  a  +  2c  is  used  in  the  numer- 
ator. Thus,  if  the  radical  sign  be  restricted  to  denote 
merely  the  arithmetical  root,  if  k  be  defined  by  the  equa- 
tion F  —  Jc-\- 1  =  0,  and  if  to  and  n  indicate  any  integers 
whatever,  equal  or  unequal,  the  value  of  x  may  be 
written. 


SIMULTANEOUS    QUADRATICS.  239 

2.  Sx'  —  bxy  +  ^y''-='^{x  +  y),  (1) 

11  :t^^  -  d>xy  +  by^  =  13(:^  +  y),  (2) 

First  Method.     Eliminate  (x  +  y). 

.'.  lO^x""  -  65:ry  +  SOy^  -  99.r'  -  72^7  +  45^/1 
.\bx'  +  1xy  —  ^y''  =  0. 
.•.(5^-3y)(a;  +  2y)^0. 
.•.  ^  =  f  y  or  — 2y. 
Substitute  these  values  for  x  in  (1), 

.-.  72y^  -:  360y,  or  45y2  -  -  9y. 
.•.y==0,  or  5,  or —J-, 
and  x^=0,  or  3,  or  f. 

Second  Method.     Take  the  sum  of  the  products  of  (1)  and 
(2)  by  arbitrary  multipliers  Ic  and  I. 

Tc(Sx^  -  bxy  +  3y0  +  ^(llo;^  -  ^xy  +  by'') 
=  {^h+l?>l)(^x  +  y).  (3) 

Determine  1c  and  Z  so  that  the  left-hand  member  of  (3)  may, 
like  the  right-hand  member,  be  a  multiple  of  a;  +  y-    This 
may  be  done  by  putting  x=^  —  y  in  (3),  from  which 
16^  +  24^  =  0. 
.'.2Tc  =  -^l. 
.•.if/^=3,  l  =  -2. 
Substituting  these  values  in  (3),  it  becomes 
2x''  +  xy  —  y'^  —-x-\-y. 
.' .  {x  +  y^i^x - y)  =^ x  +  y, 
or"  (x -{-y){2x —  y —  V)  =^0. 

:.  either  x  +  y  =^0,  ov  2x  —  y  —1=^0, 
,'.  y^=  —  x,  or  22^  —  1. 


240  SIMULTANEOUS   QUADRATICS. 

Substituting  these  values  for  x  in  (1),  it  becomes 

16^2  ==0,  or  10^'-7rr  +  3=-272;-9. 
.*.  x  =  0,  or  3,  or  -|, 
and  3/  =  0,  or  5,  or  —  -|-. 


3. 


x^  +  xy  +  7/'  =  a'  +  ah  +  b\  (2) 


(a'  +  by-a'b' 

a^y  +  xi/^       _       a^b  +  ab^ 
{x^  +  yy  -  ^y  ~  («'  +  ^')'  -  «'^' 


Write  z  for 

a;^  +  y'^                     ci^  +  b^ 

(3). 

.       2              y^ 

l-z'      l~¥ 

\z  =  k  or  —  -• 
fc 

xy             ab           c?  +  b^ 

\     •  ^      =— —  or  !— — . 

(3) 


x"^  +  y^      d^  +  b'^         —  ab 

xy                      ab                    a^  +  b'^ 
.*. sz = or  ■ 

x'^  +  xy  +  3/^      a^  +  ab-\-  b^        a^  —  ab  +  b'^ 

(2),  .-.  xy  =  aJ  or  (a'  +  J^  "'^  +  ''^  +  ^[-  (4) 

a  —  ab  -}-  0 

V[(2)  +  (4)], 

\a^  +  ab  +  b^ 


^(2a^^ab  +  2b^)^^ 


ab  +  b' 


SIMULTANEOUS    QUADRATICS.  241 

V[(2)-3(4)], 

and  X  —  7/  =  ±:(a  —  b), 

''  +  ab  +  b' 


i^(2a'  +  ab+2b')J^ 


-ab  +  b' 

.'.  X  =  ±:a,±:  bj 

or  i[-y/(2a'-ab  +  b') 

3/  =  ±  5,  =b  a, 
or  -.       i[^(2a'-ab  +  b') 

ix  +  i/)(x*  +  ^')  =  b.  (2) 

Put  2=^^- 

x'  +  f 
.    1  —  z       a 


1-22^     b 
.■.2az'-bz-(a-b)  =  0. 
:.4taz  =  b±  y/{%a? -  Sab  +  b'')  =  b +r,  say. 
._x^_^b±r  .g. 

_./g  +  y__|_    I2ffl  +  6  +  r 
x  —  y  \2a  —  6  —  r 

.  a;  _  V(2a  +  &  +  r)  +  V(2 «  -  6  -  r) 
■>      V(2«  +  6  +  »-)-V(2«-&-'-) 

_  r  V(2a  +  &  +  r)  +  V(2a  -  5  -  r)?     .4. 
2(6  +  r)  ^^ 


242  SIMULTANEOUS    QUADRATICS. 

/OX  /     N5/4a+2Z)+2rV4aV/4a-Z>-rV       , 

.     .0 fy\p2a^(2a+b+r)(Aa-b-ry-\ _   , 

\i/J[32(2a  +  b  +  r)(4:a-b-  rfj 

/.x  _\^(2a  +  b  +  r)  +  ^(2a-b-r)r 

^  ^'  ^  1024(2a  +  b  +  r)(4a  -  ^  -  r)^ 

.  V(2a  +  ^  +  r)  +  V(2a-^>-r) 

2^[(2a  +  ^»  +  r)(4a-6-r/]  ' 

in  which      r  =  ± V(^  a^  —  8  a5  +  Z>^). 

The  value  of  y  may  be  derived  from  that  of  x  by  the  first 
form  in  (4). 

5.  x^  =  ax~  by,  (1) 

y'  =  ay-  bx.  (2) 

^X(l)-yX(2), 

yX(l)-xX{2), 

xy  {x^  —  'if)=^b{x'^  —  y'^). 

.*.  either  x  —  y  =  Qy 
from  which  :r  =  3/  =  0,  or  -^/(a  —  ^),  (3) 

or  :r*  — r^V  +  ^V  +  ^Z  +  y^'^^C^  +  y)'  (4) 

and  xy  {x"  +  xy  +  3/')  =  b{x  +  y).  (5) 

(4)  +  (5),         {X  +  3/)X^^  +  3/^)  ^{a  +  b){x  +  y).  (6) 

(5),  (^  +  yy  -  {x'^  +  ff  =  ^b{x  +  y).  (7) 

V[a)^  +  4(6)], 

(^  +  yy  +  (o;^  +  yy  '-=2t{x  +  y\  (8) 

in  which      t  =  y/[(a  +  by  +  4: b'].  (9) 


I 


SIMULTANEOUS    QUADRATICS.  243 

i[(7)  +  (8)], 

.■.ix  +  yy  =  {2b^t){x  +  y). 

:.{x  +  yf  =  2h  +  t. 

.■.{x  +  y)  =  ^{2b  +  t).  (10) 

(6) -^(10)    :.x'  +  y''=     ^t^     •  (H) 

^  '      "-     '  ^y       -^(26  +  0  ^     ^ 

2(ll)-(10)^ 

2a  — i! 


^(25  +  0 
,.^_         V(2«-0. 

y  ^(26+0 

(10)  and  a;  +  y  =  Yl^f  +  ^X 

^     '  -^      ^(26  +  0 

.  ^_V(2&  +  ^)  +  V(2a-<) 
^(26  +  0 

and  .^V(2^  +  0-V(2a-0^ 

in  which      <  ==  V(«'  +  2a6  +  5 6'). 

6.  a;*  — c*  =  w(a;+y)*,  (1) 

y*  +  e*  =  n(x-yy.  (2) 


Let 


x~-y 


.•.2  +  l=_^and2-l=-?^.  (3) 

a;-y  a;-  y 

(l)  +  (2),         x'  +  y*  =  m(x  +  yy  +  n{x-yy. 

.:  (z  +  1)*  +  (z-  ly  =  16  (mz*  +  w). 

.-.  (8m-l)2*-62^  +  (8n-l)  =  0. 


244 


SIMULTANEOUS    QUADRATICS. 


.  _     /3+V[9-(8m-l)(8n-l)1 


•    z=       /3-r  VI  o  —  yuv/^—  xyi^u/i.—  -l;|  ... 

\  8m -1  ■        ^' 


(2)  and  (3),     (z  -  !)*(»  -  y)*  +  16  c*  =  16  w  (a;  -  y)^ 

2c 


and 


and 


^[16«-(z-l)^] 

2cz 
>[16n-(2-iy]' 

c(z+l)  _  c(z+l) 


(5) 


y=T 


^[16w-  (z  -  1/]      ■</[(2+l)*-16mz*]' 
g(^-l) 


^[16«-(^-l)*] 
and  the  value  of  z  is  given  by  (4). 

7.  x''  +  ^'  =  i(27n  +  n''), 

x^-{-i/  =  mn. 

•*.  (^  +  yy  -  "^xy  =  i(2m  +  n'), 
and  (a;  +  y)^  —  3  xi/  (x -{-]/)  =  mn. 

Let  t^  =  07  +  2/  and  V  —  rpy,  and  the  equations  become 

u^  —  Suv^=  mn. 
EHminate  v, 

•.  ^6^  —  (2 m  +  n^) ?^  +  2 "inn  =  0. 

•.  1^*  —  (2m  +  7i^)  2^^  +  2mni^  =  0. 
.'.  u^  —  27nu'^  +  771^  —  7iV  —  27nnu  +  m^. 
.'.  t6^  ~'m  — db  (?^^6  —  m). 

(the  value  u=0  was  introduced  by  the  multiplication  by  u), 
or  w^  +  Tii^  —  2  m  =  0. 

.-.  u=:^^[—7i±:  ^(n^  +  8m)]. 


SIMULTANEOUS    QUADRATICS.  245 

I.\v  =^(71'  — m), 
or  iln'  +  8m  q=  3^  V(^'  +  8m)]. 

.'.  u  and  V  are  completely  determined. 
Also,  x-{-y  =  u,  X —  i/  =  ^(u^ —  4iv). 

If  m  =  7  and  n  =  5,  the  above  equations  become 
I  x'  +  y'=lS,2indix'  +  f  =  S5. 

Solving,  as  above,  gives 

I  2^  =  5,  or  2,  or  —  7  ; 

2^-12,  or -9,  or  36. 
.•.  r^  +  y  =  5,  or  2,  or  —  7  ; 

^  —  y  =  db  1,  or  zb  V22,  or*±:  i  V23. 
.-.^  =  3,  2,  i(2±V22),  or  i(-1±:i^2S); 
3/ =  2,  3,  i(2=FV22),  or  i(-7=F^  V23). 

8.  o,^+y  =  XZ 

^  +  2/'  ==  I- 

Testing  this  for  rational  linear  factors,  it  is  easily  reduced  to 
(y-l)H/  +  2y  +  i)  =  0. 

.•.y  =  l,  or|(-2±V2); 
«  =  i,  ori(-ld=4V2). 

9.  (2x-'!/  +  z)(x  +  y  +  z)  =  9,  (1) 
(a;+22/-z)(a;  +  2/  +  z)  =  l,  (2) 
(a;  +  y-22)(a;  +  y  +  2)  =  4.                      (3) 

Let  s_=  a;  +  y  +  2,  and  the  equations  may  be  written 


246  SIMULTANEOUS    QUADRATICS. 


(s  +  a;-2y)s  =  9, 

(4) 

(s  +  2/-22)s=l, 

(5) 

(s-32)s  =  4. 

(6) 

(4)  +  3(5), 

(4s  +  a;  +  2/-6«)s=12, 

or 

(5s-7z)s  =  12. 

(7) 

3  (7) -7  (6), 

[(15s -  21z)  -  (7s  -  2l2)]s  =  8. 
■.8s'  =  8.     .•.s  =  ±l. 

Substituting 

in  (1),  (2),  and  (3),  they  become 
2a;  — y  +  z  =  ±9,  a;  +  2y  — 2  =  ±1, 
«  +  y— 2z  =  ±4. 

■.  a;  =  ±4,  y  =  =F2,  2  =  =f1. 

10.  x^  -{-y^  =  a, 

u'  +  v^  =  b,       ' 

xy  -\-uv  =  <?, 

xu-^-yv  =e. 
Let  t^=^  xy  —  uv. 

.'.(x  +  yy--=a  +  c  +  t, 

{x~yy~a  —  c~t 

(^^  —  -y)^  =  5  —  c  +  i5. 

Also,  2  (:?;2^+y?;)  =  (a;+3/)(t6+t')  +  (:r— 3/)(w— -y)  :=2^. 

+  ■yj\{a-c~t){h  -  ^  +  0]  ==  2c. 
=.\^e\a-  c  —  t){h  -  c  -\- 1), 


SIMULTANEOUS   QUADRATICS.  247 

.'.[(a-  by  +  ^e']f-2  (a'  -  b')  ct 

+  {a-\-byG^  -  ^e\ab  +  c^)■\-^.e'  =  0. 

.  (o?~b'')c±:2c-sJ\{ab-e^)\{a~by -  ^c" -  e^yW 

{a-by+^^ 


llo 

xy  =  u%\ 

(1) 

x  +  y  +  u  +  v=^a, 

(2) 

a^  +  f  +  u'  +  v^^  b\ 

(3) 

x^  +  f  +  w'  +  v'^^cK 

(4) 

Let  x  +  y= 

--i{a  +  z). 

,\u  +  v  =  i{a  —  z). 

(5) 

Also,  let  r  - 

-  xy  =  uv. 

(x  +  yf  =  x'  +  y'  +  Sxy(x  +  y), 

(6) 

(u  +  vy  =  ii  +  -y^  +  3  uv  {u  +  v). 
.•.a(3^^  +  a^)-4(6^  +  3ar).  (7) 

Also,  (^+y)^=:^^+3/^+5rry(:^^+2/'^)  +  10a;y(:?;+y), 

.-.  ti(52^+10aV+aO-16(c^+55V+10ar^).    (8) 

Eliminating  r  between  (7)  and  (8), 

45  a'^2*  -  30  a  (a^  +  2  5^)  2^  +  a'  -  20  a^b^ 
-805«+144ac^-.0. 

.\15a^^-5(a^  +  2^0 

-  ±  2V[5(a^  +  557-  180 ac^.  (9) 

._^_     /a^+2^^±2V?^r(c^^  +  5&y-36a^ni  (iq) 

*  "         \  3a 

(7)  and  (9),     12ar- a^-4^^  +  3a2^ 

-2a^-25^±2V^i[(^'+5Z^0'-36ac^]|. 

_  5  (g^ -  Z)^)  ±  Vr 5  (g-^ + 5  Wy -  180 a&^ .  ^^v 

••"^  30a  ^     ^ 


248  SIMULTANEOUS    QUADRATICS. 

(10)  and  (11)  give  the  values  of  z  and  r,  which  may  now  be 
treated  as  known  in  (5)  and  (6). 

^  +  y  =  2*(a  +  2;),  and  ^3/ =  r. 
:,x-y^\-yJ\{a-\-zJ-\^r\ 
:,x=\\a-\-z  ±  V[(«^  +  ^y  -  16r]  \ , 
y=\\a-^z  =F  V[(^  +  2;)'  -  16  r] } . 
The  values  of  u  and  v  may  be  obtained  from  those  of  x  and 
y,  respectively,  by  changing  z  into  —z. 


12. 

ax  —  by  ^=^  cz  ^=^  - -\ — 

.    a  _b  __  c 
yz      zx      xy 

(1) 

{\)-^xyz, 

a  +  h-\-c 
xy  +  yz  +  zx 

(2) 

Also  from 

(1)  -^  :?;3/2;, 

a        1/1.1.  l\_xy  +  yz  +  zx 
yz      xyz\x  ^  y  ^   z)            x^y'^z^ 

(3) 

(2)  X  (3), 

0?  __a  +  h-\-c 
,'.  aV  =a~{-b-{-c. 

13. 

y~}-z  —  x_z~}-x  —  y_x  +  y  —  z 

(1) 

a                    b                    G 

xyz  =  TT}^. 

(2) 

(1). 

Z                 X                 V            Wj 

— ^^—  r=— :lL-.  =  — ^— =  — ,  suppose; 
a-\-b      b-\-c      c+  a      r 

(3) 

then 

xyz                 __  m^ 

{a  +  b){b  +  c){c  +  a)      r' 
:^r^=:(^a  +  b)(b  +  c){c  +  a). 
Hence  the  value  of  r  is  known,  and  from  (3) 
rx='m(b-\-  c). 


SIMULTANEOUS    QUADRATICS.  249 

14.  y-\-z  =  '2.axyz,  (1) 
z  -{-x^^^  hxyzy  (2) 
x-[-y=2  cxyz.                                                (3) 

^         2a  2b  2c        a  +  h  +  c 

=  -—^-^  = ^ ^ ? /4^ 

6  +  (?  —  a      c  +  a+b      a  +  b  —  c     ^  ^ 

.-.  a^yh^  ^  — ^^^ 

{b  +  c  —  a){c-\-a  —  b){a-{-b~c) 

"^y^  ~  (hj^c-a){c  +  a~b){a  +  b-c) 
Hence  the  value  oi  xVz^  is  known,  call  it  -,  and  substitute 
in  (4)  1  =  - ^ 

.\rx  =  b  -\-  c  —  a, 
in  which      r"^  =  (b  -\-  c  —  a){c  +  a  —  b)(a  -{-b  —  c). 

15.  3/^  +  ^2  _^(^_^^)_^^  (1) 

2^+a;^-3/(2;+^)  =  &,  (2) 

a7^  +  3/^-z(r^+y)-e.  (3) 

(l)  +  (2)  +  (3), 

2{x'^  +  y"^  +  z^  —  xy  —  yz  —  zx)  =  a-{-b  +  c.    (4) 

(1)  may  be  written 

x^  +y''  +  z"  -  x{x  +  y  +  z)  ^  a.  (6) 

(2)  may  be  written 

^''  +  f  +  z'~y{x  +  y  +  z)  =  b.  (6) 

(3)  may  be  written     * 

x'  +  f  +  z^-z{x  +  y  +  z)  =  c.  (7) 


250  SIMULTANEOUS    QUADRATICS. 


,         ,  a  — u       u  —  c       c  —  a 

/.x  +  ^  +  z-= = -= 

^  —  X     z  —  y     X  —  z 

.  (r  4-  V  4- ,)'  -  (« "-  ^y + (^  -  '^y + ('^ " «)' 

_o}  -{- y^  +  c^  —  ah  —  he  —  ca 
^^  +  y^  +  ^^  —  xy  —  yz  —  zx 

(A\  —  2  {a^-\-h'^+c^—ah—hc—cd^        /gx 

^  ^  ~  a  +  h  +  c  ^  ^ 


(9) 


^2(a^  +  ^'  +  g'-3aZ>g) 
{a  +  h  +  cy 

Write  r^  for  2  (a'  +  ^'  +  ^'  -  3  ahc). 

(9)  ...:,  + 3/ +  ,^_^.^.  (10) 

Returning  to  (8), 

.     ,      ,    X2     2(a?-\-h'^+c^—ah—hc—cd)         /qn 
a+6  +  c 

(4)  2Cr^+y-+.^-rry-y.-.^-)  =.  ^'^+^+'^'-       (11) 
i[(8)  +  (ll)],a;^  +  y^  +  z'  =  '^'  +  f  +  <  (12) 

(5)  and  (10),  x'  +  f  +  z' ^f—  =  a. 

a-\-  b-\-  c 

:.  rx^{a+h  +  c)  {x^  +  y''  +  z^)  -a(a  +  h  +  c) 

(12)  =  o?  +  h''  +  c"  -  o.{a  +  h  +  c) 

^y^j^c'-aih  +  c), 

(5),  (6),  and  (7)  are  symmetrical  with  respect  to  {xyz\ahc) ; 
(10)  shows  this  substitution  does  not  aiFect  r,  and  conse- 
quently the  values  of  y  and  z  may  *be  written  down  at 
once  from  that  of  x. 


SIMULTANEOUS    QUADRATICS. 


251 


Ex.  62. 

1.  ^[{1  -xy  +  f]  =  l^{l  -x)y',   x'  +  4.y  =  f  +  4., 

2.  I0x'-^y''  =  2x^',    8^2-63/^  =  130;. 

3.  xy^(^^-xy  =  {2-y)\ 

4.  ri:'  +  y2=-8:r  +  9y=144. 

5.  x'  +  f  =  x  +  y  +  l2-    xy  +  ^  =  2{x  +  y). 

6.  x  +  xy  +  y  =  b)    x'  +  xy  +  y''^1, 

7.  x^  +  y^^1xy  =  2^{x  +  y). 
Q  2  ,         ,2  35         28 

X  +  3/       ^ry 

9.  x'  +  xY  +  y'=^l?>^\    x^y  +  xY  +  xif  =  11^. 

10.  (rc  +  y)(:r'+3/0  =  17^y;  {x-y){x^ -y'')  =  %xy. 

11.  25(:^^  +  y^)  =  7(^  +  3//-175a;y. 

12.  2a;2  -  3/2  =  14  {x''  -  2^^)  =  14  (r^  -  3/). 

13.  2x'-Zxy=:^^{x-^y)]  ?>{x'~?>y'')  =  2{2x^-~^xy). 

14.  2:r2-:ry+5y2=10(a;+y);  x'+^xy+'^y'  =^l^x+y). 

15.  (2a;-3y)(3:?:  +  4y)-=39(:r-2y); 
(3 :^  +  2y)  (4  :r  -  3  y)  -:  99  (:?;  -  23/). 

16.  {x+2y){x+?>y)  =  ?>{x+y)',    {2x+y){^x+y)  =  2^(x+y) . 

17.  r?;  +  3/-=8;    :r*  +  3/*  =  706. 

18.  :?;  +  3/  =  5;    0:^  +  3/^  =  275. 

19.  :z;  +  y  =  2;    13(a;^ +  y^)  =  121  (:?;^  +  3/=^). 

20.  rrr  +  y  =  4;    41  (rz:^ +  3/')  =  122(a;*  +  y*). 

21.  r^2__  5^y  _|_y2  _^  5  ._  Q  .    xy  =^  x-\-y  —1. 

22.  :?;2  _|_  y  —  5  (^.^  __  y^  .    ^^  +  3/^  =  2 (a;  —  3/). 


252  SIMULTANEOUS    QUADRATICS. 


23.  ?>{x'  +  y)  =  ?>{x  +  y'')  =  l?>xy. 

24.  10  {x"  +  y)  =  lO{x  +  y^)  --  13  (x"  +  y% 

25.  x^  +  y  =  ^',    ^  +  y2^2^. 

26.  9(:r^  +  3/)-=3(ri7  +  y2)  =  7. 

27.  ^  +  ^y  +  y'=5;  r?;*'^  +  ^y  +  y^  =  17. 

28.  :?;  +  3/  =  2;    (:r  +  1)^  + (y  ~  2/ -=  211. 

-■  ^(-i)&+i)=*wi)(^i):  |g±-;=i@^)- 

30.  a;  +  y~  —  ;    x  —  y  ^=  xy. 

xy 

31.  a;  +  3/  +  l--0;    :r^  +  /  +  2--0. 

32.  x  +  y^l;    3  (;r' +  y')  =- 7. 

33.  4.t/==5(5-.t);    2(^^  +  y')-=5. 

34.  27^77 -IT;    9(:r^  +  3/^)=:~8. 

35.  (.1-  +  yj  +  ^xY  =  5  -  12y  ;    y  {x"  +  y^)  _^  3  =,  q. 

36.  X  -{-y  =^xy\    x^ -{- y"^  =^  x^  -\-  'jf. 

37.  </+3i/-l)  =  2y'+23/+3  ;  y(:r^+3r^-l) -- 2a;2+2a;+3. 

a^      b^  a      b         \a      bj 

39.    a;3  +  rr2/^  =  a;    '}/ -\~x^y^b. 


40.    a;  +  3/  =  <^; 


^  —  . 


-2/ 


41.  x'^  +  ay"^  =  cL-r    .   ^^2 _[_ y2  ^  (<2^ _  i)y^ 

42.  x  +  y^  =  ax]   x^  -\-y  —  by. 

43.  x-\-y'^  ^^  ay'^ ;    rr'^  +  3/  —  ^^^. 

44.  a;*— y*  — a^(a;— 2/)^ ;    x^  —  x'^y-\-xy'^  —  'i^^=b'^(x-\-y). 


SIMULTANEOUS    QUADRATICS.  253 

45.  (x  +  y)  {pc^  +  Sy^)  =  m ;    ix  —  y)  {x^  +  3y^)  =  n. 

46.  x'^y'^  =^y{a  — xy  ^=x(b —  yf, 

47.  x^  (^  —  y)  =  y^  (a~  x)  ~  (a  —  xy(b  —  y)^. 

i     48.  a\x''  +  e')  =  b''(x  +  yy;    a' (y' +  e')  =  c' (x  +  yf. 

49.  x^  —  y^  =  a  {x'^  —  y'^)',   x^  +  y'^  =^h{x  +  y). 

50.  x-\-y  =  a',   x^  +  y^ ^=hxy. 

51.  x  +  y  =  xy^=x^-\-y'^. 

52.  x  —  y^==^~^=x'^  —  y^, 

^    y         ^ 

54.  ^' + ^y + 3/'  ^  ^' + 3/'  ^  ^y . 

x^  ~xy  •\-y'^  a  h 

55.  rgV  +  ^y'==^2^    2>    oc'y  +  xy'  =  h. 

56.  x^y-\-xy'^  =  a{x^-\-y'^)\   x'^y~xy'^~h{x^-~y'^), 

58.  r^^  +  y2  —  ^^2^2  __  ^^  ^^  _|_  ^^^ 

59.  a^^y  =  a  (^^  +  3/^)  —  Z>  (a;  +  3/)^. 

60.  rry  (:?;  - [-  y)  —  a  ;    r?;^  (x^  +  ?/^)  =  Z). 

62.  ^r*  +  3/*  =  m (r?;2  +  y"^)  ;    o;^  +  a-y  +  y^  =  92. 

63.  a^>(rr  +  y)--^y(a  +  ^>);    r^' +  y^ -=  a' +  5^ 

64.  x^  +  f  =  a{x  +  y)]    x^ ^y""  =  h{x  +  yy. 

65.  :^'  +  3/2-:a;    ^^  +  y^ -- 5  (a;' +  3/^). 

66.  xy^=a\    x^ -{-y^  ^=h{x^ -^y^). 


254 


SIMULTANEOUS    QUADRATICS. 


67.  {x~y){x'  +  f)  =  {a-h){a^  +  ¥)',   a^-y'^  =  a'~h\ 

68.  x^  —  y^  =  a\    x^  +  i/ =.h{x —  y). 

69.  x  +  y^a\    x^  +  y^-=h. 

70.  x  +  y  =  a\   x'-\-y'='b. 

71.  x-\-y  =.a\   x^  +  y"^  ^=  SVy^. 

72.  x-\-y=-a  +  h]    {a-hy{x'  +  y')  =  {x~yy{a}  +  h'), 

73.  x  +  y  =  a',    c{x^ +  y^)^ xy{x^ +  y^), 
14.,  {x  +  yy  =  a{x''  +  y'')\   xy  =  c{x  +  y). 

75 .  x^y  +  xy"^  =  a^ ;    c^  (x^  +  y^)  =  x^y^. 

76.  r?;^  =  a (x^  +  y^)  —  cjt^/  ;   y^  -=  c  (x^  +  y^)  —  axy. 

78.  o;'^  —  2/*  =  a^^y  ;    (:r^  +  y'^y  ==  ^^  (r^;^  —  3/^). 

79.  (a;  +  y) x^y'^  —  a;    x^  -{-y^  =  h. 

80.  (a;  +  y)  ^y  =  <^  ;    x^  -{-y^  =^  hxy. 

81.  a;*  +  y*  =  a(a;  +  y)';    r?;^  +  y^  =r  ^ (:r  +  y)l 

82.  x^  +  ii^^y'^  +  y*  "^  <^ ;  ^^  —  ^y  +  y^  —  1. 


77.    x"^  —  y'^-=^a^\   x^  —  'i^  =  c^ 


-y 


,.   x\l  +  xY)_ari+xy'\ 
y\l  +  xyy      b\l-xy] 

x'  +  y''=.a{x  +  y)\    x' +  y^  =  h{x^ +  y'). 

x^  +  y^^a]    {x  +  y){x^  +  y')  =  b{x''  +  y''). 

{x"  +  y')  {o(^  +  y')  =  axy  ;    (a;  +  y)  (^*  +  y*)  =  ^>rry. 

(^  +  y)^(a;^  +  y^)  -  a  ;    (:r^  +  f)\x'  +  yO  -  b. 

(x  —  y)  (:?;'^  —  y^)  (ri;*  —  y*)  —  4  aa:y  ; 
{x  +  y)  (o;^  +  yO  (o;^  +  y^)  =  5(^  -  y). 

x'^y  +  ^y*  =  <^  {oc^y  +  ^y^)  =  5  (rr*  +  y*). 


83.    {x'  +  f)xy-- 

84. 
85. 
86. 
87. 
88. 

89. 


SIMULTANEOUS   QUADRATICS.  255 

90.    a{r'  +  y^)  =  ah{x  +  y)=^hxy{x' +  y^). 


91. 


X^  _  y3  ^3  _  ^3  ^      X"  —  ^ 


92     x^  +  f _a^  +  h\    x'  +  y' __a'+b'' 
x^  —  ?/^      a^  ~  h^ '    x^  —  3/^      o?  —  h^ 

93.  x^  ^^2ax  —  hy  ]   3/^  —  2 ay  —  5^. 

94.  {x  +  y)  (^^  -f-  y^)  =  a  ;    {x  —  y)  (o;^  —  y^)  —  b. 

95.  (^+y)'(^'+^y+y')  _  3^2 . 


(x'+y')(x''+xy+y') 

96.  (:?;  +  y)  (^^  +  y^)  =  aa;y ;  (x  — y)(ci^ —  y^)  =  hxy. 

97.  (:^+y)(^^+y^)  =  a(:r^+y^);    (a;-y)(^^-y^)  ^  ^(:r^+y^). 

98.  (^+3/)'(^+y')      _  ^2 .       (^-y)X^ -/)     _.  ^2 
(x'+xy+y^Xx'^+y')  '    (a;''^~^y+y'')(a;'+y') 

99.  (^'+3/')(^+y)'_,o^2.  (^'-f)(^-yy_2b' 

x'^  +  xy  +  y''  '        x'  —  xy  +  y'' 

100.  (^'  +  3/')(^  +  3/y_3^2.    (^'-3/')(^-yy_352 

(:r'  +  rz:y  +  y'7  '     {x"  -  xy -\- y'-J 

101.  a:y(a;  +  y)(^'  +  y')  =  o^;   xy{x  —  y){x^-f)  =  b. 

102.  :^(^+y)(^+2y)(^  +  3y)-a^  (:^.+y)2+(a;+2y)^=  ^. 

103.  (:r+l)(y-l)  =  a(:r-l)(y  +  l); 
(^^  +  l)(y-l/^^^(:.-l)Xy^  +  l). 

104.  x  +  y  =  a(l+xy)',    {x  +  yf  =-¥{!+ xY). 

105.  a:  +  y==a(l  +  a;y);    ^^  +  y^  =  Z)'(l +ri;y). 

106.  (a;  +  y)(y-l)-a(:r-l)(y+l); 
{:^-V){y-l)^b\f-l){x-l). 


256  SIMULTANEOUS    QUADRATICS. 

(1  +  ^)(1+.V)^        {l  +  x)\l  +  yr^ 
•    {l-x)(l-y)         '  (l-^)(l-3/)        • 

108.  (^  +  ^)(g  +  y)_„.  (c^  +  ^0(g^  +  yO_^_ 

(c  —  «)  (o  —  y)         '  (c*  —  a;*)  (e*  —  y*) 

109.  (^  +  ^)(y  +  ^)-a.   (^'  +  mO(y*  +  wO_^ 
(a;  —  7n){y  —  n)         '    (x  —  m)\y  ~  n)* 


110. 


116. 


(a;-l)(y-l)      o'  (:>fi-l)(f-l)      c 


111.  ^Il±i^)  =  a;^!Ii+^  =  J. 
x(l  +  /)         '  x\l  +  y') 

112.  1+5==.    I?;    ^Ili5+^==5. 

x(l+y')         ^a  +  f) 


(x  +  y)(xy+l)_a'  +  h\  x(y'+l)_ 


a- 


{x  —  y){xy  —  V)       2ab    '  y(a^— 1)      a-\-b 
117     {x  +  y){l  +  xy)  ^    ,.„  _  ,^  .  a:(l-,yO  ^  , 

118.   (^  +  .y)(l  +  ^y)_„.   (x'  +  ?/)(l+a;y)_^_ 
(a;-y)(l-a;y)  '    {x'-f){l-xY) 

120    (^'+.v^)a  +  <v*)-a-  (f^±lKl±£^-5 

{x'-f){l-~xS/)         '    (x-y)(l-xy) 


SIMULTANEOUS    QUADRATICS. 


257 


{x-y){l-xy)         '    (^-2/')(l-^y) 

122     (^'  +  xy  +  f)  (1  +  xy  +  xY)  ^  ^  . 
{x  +  y)\l  +  xyy 

(x'  -  xy  +  y^)  (1  —  xy  +  xY)  ^  ^ 
{x~-y)\l-xyy 

123.  x'-^x''y  +  ba'x  +  y''  =  0-   y^ -  xy—2a''x=^0, 

124.  2x{y''-2xy  =  a]  y(Y -^xj^i^  —  ^x)  =-1. 

Hence,  deduce  the  solution  of  ^z;^  —  5  :?;^  +  2  =  0. 

125.  ^xyix'^fj^^a-,   {x^ -y''){x^ +  yy  =  b. 


Ex.  63. 


2.  ^^  — ?/2;  =  l, 
y"^  ~xz  =  2, 
z^  —  xy=^  3. 


1.    (2^  +  y-4^)(^  +  y  +  2)  =  24, 

3.  (:t'  +  2y-3^)(rr  +  y  +  ^)-2(a;y  +  2/2  +  ^:r)-:-12, 
{2x-2>y  +  z){x  +  y  +  z)+{xy  +  yz  +  zx)=^Q>l, 
{^x  —  y  +  2z){x  +  y  +  z)—b(xy  +  yz  +  zx)  =  b. 

4.  ^^-y2==0,  5.    (^'  +  3/'  +  2'/  +  (^  +  3/)'=-31, 


^'  +  y'  +  2'  =  21. 

6.  x  +  yz  =  l^,     ' 
y  +  zx=- 11, 
z-\-xy  =  10. 

7.  x  +  y  =  8z, 

x'  +  y'  =  lS4:z\ 
^  +  y'  +  2'=-134. 


8.  X  +y  =5z, 
x^  +  y''  =  S9z, 
x^  +  y'=105z\ 

9.  a;  +y  =72;, 
af  +  y'  =  25z\ 
x'  +  y'  =  674cz'. 


258  SIMULTANEOUS    QUADRATICS. 


10. 

V3/     V 

:?;^  + 2/"  =  20,2720. 

11. 

x-^y :  y^z :  2;+^ : :  a :  ^ :  c, 

(a+^+c?)  ^72;  =  2. 

12. 

x^y :  y+2; :  2;+:?:  \\a\h\c, 
(a-\-b-{-c)xyz  ■=  2  (^'+y+2:). 

13.    (a7  +  y  — 2;):r  =  a,  15.    (y  +  z)(2x  +  y  +  z)  =  a, 

(x-y  +  z)y  =  b,  (z  +  x)(x  +  2y  +  z)  =  b, 

(—x  +  y  +  z)z=-c.  (x+y)(x  +  y  +  2z)  =  c. 

16.  X  (y  -{-  z)  :  y  (z  -{-  x)  :  z  (x  -\-  y)  =  b  -j-  c  :  c  -\~  a  :  a  +  b, 
xy  -{-  yz  +  zx  =  (a  +  b  +  c)  (x  -\-  y  -\-  z). 

17.  (a  +  b)x+(b  +  c)y+(c+a)z  =  {a+b  +  c)(ix+y+z), 

(^  +  y)^  +  (y  +  .^)^  +  (^  +  ^)^==4(a^  +  ^^+c0. 

18.  c(5;  +  2/)  +  ^(^-2;)-a(y  +  2)  =  0, 
b(x  —  z)~(a  —  c)  y, 

x'^  +  y'  +  z'^a'  +  b'  +  c^ 

19.  ^  +  y  —  a2;  =  :r  —  Z>3/  +  2;  =  —  c^  +  3/  +  2;  =  xyz. 

20.  (a  +  5  +  c)  (:r  -  y)  +  a(^  +  0)  -  ^(y  +  z)  =  0, 
(a  +  ^  +  c)  (ix  —  z)  +  alx  +  y)  -  c(y  +  z)  =  0, 

(b  +  cy'^(c+ay~^(a  +  by     ' 

,  X  ,  V      7  ,  z 

21.  a:v  +  -  =  «,     2/21+^  =  6,     zx+-  =  c. 

^      z  X  y 

22.  3/  +  2: :  2;  +  ^ -^  +  3/ :  •  ^ +  ^^  ^  +  <^  •  »  +  ^, 

(^  +  y  +  2)  (0:3/2;)  =  (a  +  5  +  c)  (:ry  +  3/2;  +  zx) . 

23.  x'^  —  yz=  a,     y'^  —  xz  —  b,     z^  —  xy  ^  c. 

24.  a;^+(y-z)'  =  a^     y^+(2;-a;y-:^>^     z2  +  (a;-y)^  =  c^ 


SIMULTANEOUS    QUADRATICS.  259 

25.  x^ '\- xy  ■\- y'^  =^  a^   y'^  +  3/2;  +  2^  ^  Z>^,    2^-  -\- zx -\- x^  =  cl 

26.  X  ■\- 1^  -- 'i'  -\-  Sxyz  —  a(x  -\-y  —  z), 
^3  _  ^3  _j_  2;3  _[_  ^xyz  —  h{x  —  y^z), 

-x'-\-f-\-z'  +  Zxyz-=-c{~x-\-y  +  z). 

27.  a;  +  y  +  2a2  =  0,  34.    (:^  -  t/)^  =  a2;(^  +  y), 
a;'  +  y"-  -  25V  =0,  a;^  -  /  =  hz{x  +  y)^ 

^n  _|_  ^n  _j_  ^n  ^  C".  {x    ~  yj  =  Cz(x'  +  /). 

28. 


29. 


30. 


31. 


32. 


33.  X  (y --l)  =  a(z~l),         39.    xy  =  uv  =  a^, 

x^y^-l)  =  h\z^  -  1),  x^y-\-u-\-v  =  l, 

x'^f-X)  =  c^(/- 1).  ^^  +  y'  + 1^^  +  'y'  =  &. 


x  +  y  —  az  =  0, 
y{x^  +  f)  =  h\ 

35. 

x-y  =  a, 
u--v  =  hj 

x^  +  f  =  c\ 

xy  =  uv, 

x{y-l){z-l)  =  2a, 

x'~y^+u'-v^  =  c(a+b), 

x\f-l){z'~l)=Uz, 
x\f~l){f-l)  =  Qcz\ 

36. 

X  +y  =a, 

oc  {y—l)  =  a{z  —  l), 

x'  +  u'  =  c\ 

x\y^-l)  =  h\z'~ll 

f-^^v'=e\ 

x^{f-l)=^c\z'-l). 

x{y  —  l)  =  a{z-l), 
x\f-l)  =  h\z^~l), 
x\f-l)  =  c\z^-l). 

37. 

xy  ^=uv  =  a^, 
x-\~y  +  u-]-x  =  hy 

x\y^~l)^h\z'-ll 

38. 

xy  =  uv  ==  a^ 

x  +  y  +  u+v  =  h, 

x'{y'+l)  =  c\z'+l\ 

x'  +  y*  +  u^  +  v'^c\ 

260  SIMULTANEOUS    QUADRATICS. 

40.  xy  —  uv  =  a^,  47.    x^  +  y^  —  <^^ 
x  +  y  +  u  +  v-=h,  ^2  ^  ^2  ^  ^2^ 

\    ^     J    ^  yi^   ^     ^  ux-\-  vy  =  c^, 

41.  xy  =  uv  —  (^,  vx-{-uy  =  n^, 
X  -\-y  +  u-\-v  =  h, 

{x  +  uy  +  {y  +  vy  =  c\     48.    x  +  y  +  u  +  v  =  a, 

xy  -\-uv  =^  Jy^, 

42.  xy^=uv, 

^j^y  +  u+v=a,  ^'  +  f  =  ^\ 

x^  +  y''  +  y?  +  v'  =  h\  u^  +  v''  =  n\ 

x^  +  y^  +  u^  +  v^  =  &. 

49.    y{\-\-x'-)--=-'^x, 

x+y+u  +  v-=a 

x'  +  y'  +  u'  +  v'  =  h\  V     1      y 

4,4,4,4  4  ^(1  +^0   =2-^. 


44.    :^y  —  i^^;,  50.    x-{-y^-u-j-v  =  a, 

X  +y  +u  +v  =a,  (^^^yyj^(u  +  vy  =  h\ 


x'  +  y''  +  v}  +  v'  =  h\  ,     ,     .2  ,  /     ,    V 

x^  +  f  +  u^  +  v^  =  c\ 


{x  +  vy  +  {y  +  uy  =  e\ 


45.    xy  —  uv, 

x+y+u  +  v=a, 
x'  +  y'  +  u'  +  v'^b', 
x^  +  y^  +  u''  +  v'  =  c\ 


61. 


X 

"la- 

—  u 

y  +  ^ 

a  — 

2u 

y   - 

2h 

—  u 

z^x 

h- 

■2u 

z 

_2c- 

—  u 

46.    xy  —  uv  =  0, 

xu-\-yv  =  a', 

a;»  +  2^»  +  M»+t,'  =  (r'.  a?  +  f  +  z'  =  e'. 


SIMULTANEOUS    QUADRATICS. 


261 


52. 


=  a, 


53. 


54. 


1  +  x  +  x'' 

i  +  y  +  3/' 

l  +  x  +  f 


x+1         fx~~\ 
'      —-a 


56. 


55. 


61. 


62. 


63. 


3/  +  1 

X^  -\-X-\-\  _  yi(x_ 

y'  +  y+i' 


1 


x-\'y  __     a 
l-\~xy      b  -{-  c 

•      ^~y ^  —  ^ 

1  —  a;y  a 

57.    - — '-^-  = •  > 

1  —  xy      of-  —  a^ 

x~y  _^    2hp 
l  +  xy^b''  —  /?2' 


(l-:r)(l-y)  I- a 
(l  +  x){l-y)_,l  +  h 
(l--x)(l  +  y)      l-h 

X  -\-y  c^  —  a? 


58. 


l  +  xy 


■y^ 


2a 


l  +  xy      a^  +  a^ 

x~-y  _h''~~^' 
1-xy      b'  +  l3'' 

60.    2ax^(b  +  c~ 
2by  =  (c  -\-  a  - 

(x+y  +  zy+x' 


X  — 

•1_ 

a  — 

1 

y- 

1 

b- 

-1' 

ar'- 

-J., 

_(l_ 

-1 

x-\-y  1  -j-  xy  m 
1  —  xy  .  x  —  y  __  2b 
x  —  y       1—xy       n 

^^    y(l  +  x'') 
x{l+y') 

■ «)  (y  +  ^). 

+  f+z^  =  ^{a-  +  b'  +  c% 
64.    ^3  +  y3=-^, 


y^ 


1       ¥ 


xy  —  xy 


x  +  y 


x'^-\-xy-\-y'^  _  x^-^^f xy 

x^—xy-\-y'^         a  b 

x^  +  x'^y^  -{-y^  ^  a, 
x"^  -}- xy -{- y'^  =  b. 


65.  xy -\--  =  a(x^  —  ?/^), 
xy =^  b  (x'^  +  y^). 

66.  a^  =  a(x''+7/)  —  bxy, 

y^  =zb  (x^ + y^) — a^y- 


262  SIMULTANEOUS    QUADRATICS. 

67.  4:c(x'  +  l)=:(a  +  b)(x~y)\ 
^ciy'-l)  =  {a-b)(x-yy. 

68.  x'-n'^^(x'  +  xy  +  f)(x  +  7j), 
^^^n'  =  -^(x'  +  xy  +  f)(x  +  y). 


69. 


X  4-  xi^ 

i 2=<^^ 


y  +  ^ 


70. 


=  h. 


x  +  y 

x^  -^-1? 

— ^^==a, 
xy 

xy 

71.  ^(3/  + 2)  =  a, 
y{z-\-x)  =  l, 
z{x+y)  =  c, 

72.  {x  +  y){x  +  z)  =  a, 
{z  +  x){z  +y)^c, 

73.  x{x  +  y  +  z)-=-a  —  yz, 
y{x  +  y-{-z)  =  h~zx, 
z{x  +  y  +  z)  =  c  —  xy. 

78.    Find  the  real  roots  of  the  system  of  equations, 

x^  -\- w^  -\-  v^  =^  a^  vw  -{-  u{y  +  z)  —  he, 

w'^  +  y'^  +v?  —  h^,  wu-{-  V  (z  +  x)  =  ca, 

v^  -\-y?  +  z^  =  <?^  uv  +w{x+y)  =  ah. 


74. 

^'  —  (y  -  ^y  =  «, 

y'-(z-xy  =  b, 

z^  -ix-  yj  =  c. 

75. 

x^  +  3/^  ==  az, 

X  +y  =  hz, 

X  —y  —cz. 

76. 

11       2a 

x^  -^  f       z^' 

1      1     2h 

x^       y^       c^ 

1+1=1. 

X       y        c 

77. 

z-\ 

{x-y){z+l)  =  2a. 

{x'-f){z+iy  =  Uz. 

CHAPTEE  VIII. 

Indices  and  Surds. 
§  60.    The  general  Index-laws  are  : 


TO                  p 

-an^,, 

(1) 

TO                   p 

TO        p 

a«    9, 

(2) 

TO 

{ahyn     = 

m              TO 

an  X  ^^ 

(3) 

{a^hf= 

TO                  TO 

an  -T-  5 «, 

(4) 

TO    p 

{an),      = 

anq. 

(5) 

The  law  connecting  the  Index  and  Surd 

symbols  is 

TO 

an  =  ^(q 

;»). 

,(6) 

The  indices 

J,  J,  },  etc. 

,  are  generally 

used  to  denote 

''  either  square-root,"  "  any  of  the  cube-roots,"  "  any  one  of 
the  fourth-roots,"  etc. 

The  surd  symbols  V>  a/'  a/'  ^^^•>  ^^^  ^7  ^^^^  writers 
restricted  to  indicate  the  arithmetical  or  absolute  roots, 
sometimes  called  the  positive  roots.     Thus, 

V4  =  2,  but  4i  -  ±  2. 

Also,  v[(-2)^]-V4-=2. 

Sy27  =  3,  but  27*  -  3  or  3  ("^-i-^^^Y 

.•.8i  =  (li)-^27. 

-^16  -  2,  but  16i  =  ±  2  or  ±  2^. 

.-.  16i=:(li)-^16. 


264  INDICES    AND    SURDS. 

With   this  restriction,  the   general    connecting   formula 
would  be 

In  the  following  exercises  this  restriction  need  not  be 
observed. 


Ex.  64. 

1.  What  is  the  arithmetical  value  of  each  of  the  following: 

36*;    27*;    16^;    32*;    4*;    8*;    27*;    64*;    32*; 
64*;    81*;   (3|)*  ;    (b^f ;    (l^\y ;    (0.25)*; 
(0.027f ;   49''-^    32«-^;    8P'^? 

2.  Interpret  a-2 ;  a^ ;   cf  \    (a^)'';  a^"';  a     \  (a~^)     ; 

3.  What  is  the  arithmetical  value  of  36"*  ;  27"*;  (0.16)""^' ; 


(0.0016P;  (ip;  (^\p  ;    (^^^)"*  ;    (5,^)-* 


? 


4.  Prove  {dry  =  (ar)"^  ;    (a"^)«  =  (a«)"* ;    a""^  ^  (a"^)"^ ; 

and  express  these  theorems  in  words. 

5.  Simplify  a*  X  a*  ;    c*  X  c?*  ;    m^  X  m"^  ;    rfi  X  n~"^^  ; 

(7i)*x(2|)*x(3i)*; 

a\    c\     d\    e-^'\     .*  .    .02^^^c6^^i  .  (i^-^i 


6.    Remove  the  brackets  from 
(a«)*;    (5)"*;    (c^^ 


(a«)*;    (5)-*;    (.')"^    (^V*;    (^"¥ ;    (/"^)"^ 


INDICES    AND    SURDS.  265 

7.    Remove  the  brackets  and  simplify 

(^*-bV"*)V"¥;  #-««.-»-«« #-i)^ 

■      8.    Simplify    -x\x~''^\-xY^f;    «[(-a;)~^(-a;)-'']~*; 

9.    Determine  tlie  commensurable  and  the  surd  factors  of 
12*;   24*;    18"*;   (-81)*;    12^;    64*;    (yV)* 

L  (6f)"'. 

I  The  surd  factor  must  be  the  incommensurable  root  of 

an  integer. 

10.  Simplify   8* +  18* -50*;    72*  +  (A^)^  -  (^1^)"*  ; 

[(6  +  2*)  (6  -  2*)]* ;    (2*  +  3*)^  +  (2*  -  3*)^ ; 
(2*  +  3*)  (4*  +  9*  -  6*)  ;  (7*  -  3*)*(7*  +  3*)*  ; 
\\{a-\-x)(x^h)-f-\{a-x){x-h)-f\'- 
[a^  +  (a'  -  o^ff  X  [a*  -  (a'  -  a;')*]*. 
Express  as  surds : 

11.  a^    .^/*;    0-*;    K'^K 

12.  ri;"+i;    3/-"+§;    a'-'^    &-*^+^. 

a  TO— 3 

13.  {ax-Vf\    {x'  —  ^x-^Xy^]    {p  —  qxy-l. 

Express  with  indices : 

14.  -{/a^]    ^c^;    yx^]    ^/y"-";    -y/iax)]    ^a-\ 


266  INDICES    AND    SURDS. 

15.    ^(c^  +  lf)^    ^(a'  +  hj;    [^ia'  +  h')J;    -^[(a-h)x]; 

16.  {Jf;  (ry*;  (6~V';  (^V';  («'^)~*; 

Simplify  tlie  following,  expressing  the  results  by  both 
notations : 

17.  aXoT^  ]  a^X  a~^  ;  a^  X  a'^  ;  a  X  a~^  ;  a~^  X  -y/a  ; 


ahK'^  X  ah'K^d ;  a*^^  X  a"  W*. 


IS   ^'  -^-  y^'  :^'  i^-  il?i^. 

c(ahy  —  ac 
hc  —  c^alSf 

1  _^  3  _3  3n  3» 

a  +  a  '^  .    a^  —  a  ^  .    a"  2  —  a2      a^  +  1  +  a 


19. 


«*_«-*'    «*_«-*'    «-?  +  «?'    «+!  +  «-' 


11 

20.    Divide  x  —  y\ijxn  —  yn\ 

x^  +  a^  +  ^Hy  /  +  a^  +  a* ; 

^ + y  +  ^  -  3  a;  V^*  ^y  ^*  +  y*  +  2^ ; 

2a5  +  256?  +  2m  -  a^  -  h^  -  c^  by  a*  +  Z^*  +  el 

Ex.  66. 

1 .    Express  the  following  quantities  (i.)  as  quadratic  surds  ; 
(ii.)  as  cubic  surds ;  (iii.)  as  quartic  surds  : 

a;  3a;  2a';  c^x\  x"" )  ?/*;  a""*;   -;  mx~~p\  0.1; 
0.01;  \Xx\ 


INDICES    AND    SURDS.  267 

2.   Reduce  to  entire  surds  : 
x^X',  «-^'a;  Z>^^5^  3-{/3;  4-^2;  V^;  i-\/4;  4^9;  3^i; 


V 

7)1 


3.  Reduce  to  their  simplest  form  : 

V12;  V8;  V^O;  a/i^;  4^'0.250;  vi;  A^i;  V^i 
5^(-320);  -.t/(i-^);  v^^^  V(«'^0;  V^'\ 

V(«^);  V^"^';  -V^""^";  V^'"+';  -v/^'"-';  vK^  +  «'); 

-^(a^  +  2  a*^  +  aV)  ;   -^\{x  -^  1)  (:r^  -  1)]  ; 
^[(a^  +  2  a^  +  a:^)  {cc'  +  rr^]  ;   ^[(:r^  -  a^)\x  -  a)]  ; 
V(4.T^-8ri;^  +  4:r);   V(S^'-16^  +  8); 
^[(rc^  -  2  +  a;-^)(^*  -  2.'r^  +  1)]  ; 

|/2^-2+2^^\ .       \nx^-^x^-\-Zx\  .       //(a^-a^)H4a^" 
\ V  :^+2+ri;- V  '     \V2 7^'+ 18:^+3;  '     \V         «-^ 

4.  Compare  the  following  quantities  by  reducing  them 
to  the  same  surd  index  : 

2:  VB;  2:^9;   V^  :  a/3  ;   VIO^a/^O;  2 V2  : -^22  ; 
<(':^(r\   ^x-.-^y;   ^x'.yy-   ^x^-.^x^-    ^a:-^h:-^C) 


268  INDICES    AND    SURDS. 

5.  Reduce  to  simple  surds  with,  lowest  integral  surd 
index : 

V(»;-v/(a/^);  a/(Vc);  ^(V^');  a/(a/^')  ;  ^(^^"); 

</«/^'');  ^/(V27);  V(a/81)  ;  ^(^81);  V(«V«); 

^(aV«);  V(^a/«^);  a/C^'a/^);  ^(^VS);  V(3-n/3); 

6.  In  the  following  quantities,  combine  the  terms  involv- 
ing the  same  radical : 

3V2+5V2-7V2;  V8-V2;  ^16  +  3-^2; 

-^16 +  -^2;  ay/x  —  -^x;  a-^x  —  b-^x; 

8Va  +  5 V^  -  7 V«  +  V(4«)  -  3V(4a;)  +  4V(9a;)  ; 

^x  +  3V(2a;)  -  2V(3a;)  +  ^{4:x)  -  -y/{8x)  +  V(12a;) ; 

7 a;  -  3  Va;  +  5^a;  -  2^a;^  +  ^a;' ; 

4^(a'x)  +  2-^(Px)-S^[(a  +  byx]; 

V[(«  -  ^')'a;] + v[(« + 5)'a;]  -  v(«'a;) + v[(i  -  «)'^]  -  V^ ; 

V(a  -  &)  +  V(16  a  -  16  6)  +  V(«*'  -  ^^')  -  V[9  («  -  *)] ! 

V(a'+«'S)-V(^'  +  «^'); 

V(a'  +  2a'b  +  ah')  -  V(a'  -  2a^6  +  ab')  -  V(4a3''). 

7.  In  the  following  quantities,  perform,  as  far  as  possible, 
the  indicated  multiplications  and  divisions,  expressing  the 
results  in  their  simplest  forms : 

V2XV6;  V3XV12;  V14XV35XV10;  V«xV(3«); 

■^aX^a'Xy/b;  V«  X  JQ  ;  V«xJQ; 


INDICES   AND   SURDS. 


269 


■^/&aX^ 


6a 


;  ^a"+'X-y/a''+';    -^b^+'x -^P''+^; 


a^-V*""';  a^^V*""";  («+«)^V(«  +  «); 

(«^_^^)^^(a_a;);    (^^  -  1) -^  ^(a;  +  1)^ ; 

(3  V8  -  5  V2  +  V18  +  V32  +  V72  -  2  V50)  X  ■y/2 ; 

(7  V2  -  5  V6  -  3  V8  +  4  V20)  ( V18) ; 

(V5  +  V3)(V5-V3);  (V2  +  i)(V6-V3); 

(3  -  V2)  (2  +  3  V2)  ;    (5  V3  +  V6)  (5  V2  -  2)  ; 

( V«  -  V^)  ( V« + V^) ;  (« V^  +  ^V«)  (^ V«  -  aV^) ; 
[  V(:r  + 1)  +v(^  - 1)]  [  V(^  + 1)  -  V(^  - 1)] ; 

[V(3a-&)  +  V(3^-«)][V(3a-S)-V(3S-a)];^ 

V(« + V^)  X  V(«  -  V^) ;  V(  V^ + Vy)  X  v(  V*  -  Vy) ; 
V[a + V(«'  -  *0]  X  V[«  -  V(«'  -  ^')] ; 
^[^  -  ^(x^  -i)^x^[x  +  v(^^  - 1)] ; 

^[a  V«  -  VC*"  -  ^'')]  X  a/[  V(«'  -  ^)  +  « V«] ; 

-;y(8  +  3V7)xv(8-3v7);  (V^  +  V^)';  (V«+V^)'; 

(a-cVa;)';  (V«^  +  VO'; 

[a  +  V(l  -  «')]' ;    [^(a  +  b-x)-^(a-b  +  x)J  ; 

S  V[(«  +  a;)  (^  +  J)]  +  V[(«  -  ^)  (*■  -  5)]r ; 

W(f5|)-^e5!)|  v(.-i)V(:4i)^ 

(^a+^5)';  [V(V«  +  V^)  +  V(V«-V^)?; 


270  INDICES    AND    SUEDS. 

[V(Vio  +  i)-V(Vio-i)?; 

\  v[a  +  V(«'  -  *^)]  +  V[«  -  V(«'  -  ^0]  r ; 

(V«+V^+Vc)(V^+Vc-V«)(V«+V«-V^) 


(Vl+^l+OW^Vf-0 


8.  Find  rationalizing  multipliers  for  the  following  expres- 
sions, and  also  the  products  of  multiplication  by  these : 

V(a  -  ^)  -  V(«  +  x) ;  V(«'  +  V'^) + V(«'  -  V^) ; 

V[8+V(24+V5)]-V[8+V(24-V5)];  ^a+^b+y/e- 

3+V2+V7;  V6+V5-V3-V2;  V«+VHVc+V^; 
V(i+«)-V(i-a)+V(i+^)-v(i-^);  a/«+a/<^; 

^a^--^c'-  ^a+^c;  V«--v^^;  A/a+V«;  V^+^y") 
^x  +  l  +  ^x-';  ^iab-')—^(a-'b);    ^2+^3-^5; 
^a+-^b  +  -^o;   a  +  ^b  +  ^c. 

9.  Rationalize  the  divisors  and  the  denominators  in  the 
following,  and  reduce  the  results  to  their  simplest  form : 

1^(2-V3);   3^(3  +  V6);   5^(V2  +  V7); 

(V3  +  V2)^(V3-V2);  (7V5  +  5V7)^(V5  +  V7); 


INDICES    AND    SURDS.  271 

a-^{-y/a-\-a);    {x —  a) -^  (-yjx  —  ^a)  ; 
(a'+ab  +  b')  ^  [a  +  -^(ab)  +  b]  ;    (x+a)  ~(^x  +  -^a)  ■ 
a-^/x  +  b^y,  2V6  1  +  3V2-2V3. 

cy/x-ey/y'    V2  +  V3-V5'     V2  +  V3  +  V6' 

V'6-V5-V3  +  V2.    2 . 

V6  +  V5-V3-V2'    V(a+1)-V(«-1)' 

2c .    «  +  .r  +  V(a'  +  a;^). 

VCa  +  c)  +  V(« "~  '^) '    «  +  a;  -  VC*"  +  «'')' 

y/{a  +  x)+  yj{a~-x)  _    1 . 

V(a  +  x)-  V(a  -  a^)  '    « V(l  +  ^')  +  6  V(l  +  a') ' 
«V(1  -  y)  -  5V(1  -  g^) .    aV(l-a')  +  cV(l-c'). 

V(i-60  +  V(i-«')  '  aV(i-<;0  +  cV(i-«')' 

vrd +«)(!+ 5)1  -  vr(i  -«)(!-  ^)1 . 

V[(l  +  a)  (1  +  6)]  +  V[(l  -  «)  (1  -  ^)]  ' 
(g  -  x)^(b'  +  f)  -(b-  y)  V(«'  +  a;^) . 
(g  +  x)-y/{b-^  +  /)  +  (6  +  y)  V(«^  +  ^0  ' 

V(l  +  g)-V(l-g)  +  V(l  +  5)-V(l-5). 
V(l  +  «)  +  v(i  - «)  +  V(l  +  ^)  +  V(i  -  f>) ' 

-^(x  +  g)  -  VCar  -  g)  -  V(a;  +  &)  +  V(a^  -  b)  . 
V(a;  +  g)  +  V(«-  «)  +  V(*  +  *)  +  V(^-  ^)  ' 
yg  ,  ^  .      |/a  +  a;\  _    l/g  —  x\  .      jg+Va; . 

|Va;-V.y.        |a  +  V(g^-l). 

1       1      V^_V^ 

_L  .  _L'  v^+Vf* 
V^    Vy    V^    V^ 


272  INDICES   AND    SURDS. 

10.    Find  the  values  of  the   following  expressions  for 
n  =:  1,  2,  3,  4,  5,  respectively  : 


1 


•(2  +  V6)''+^-(2+V6)  _  (2-V6)''+'-(2-V6)" 


2V6L  1  +  V6  1-V6  J 

11.  Show  that 

2(^^sb+v(^-i)r-'+b-v(^'-i)r-'=F2] 

is  a  square  for  n  ^  1,  2,  or  3,  respectively. 

12.  Extract  the  square  roots  of: 

^  +  y  —  2-y/(a;y)  ;    a-}- c-]-e-{-2-y/(ac -{-ce); 
a  +  2c  +  e  +  2 V[(a  +  c){c  +  e)]',   2a  +  2-yJ{p?  -  c^)  ; 
2[a2  +  h''-  V(a*  +  a'^'  +  h')] ;   x-2  +  x-^', 
V^+2  +  V^"';   x  +  ^x^  +  x^  +  2x^x  +  2x'^x] 
x^—xy  +  \y''+^{^a^y~SxY  +  xy^)', 
2x+^{Zx^+2xy-y'')\  5-2V6;  10  +  2V21;  9  +  4V5; 
12-5V6;  70  +  3V451;  4-V15;  4-V15;  7+4V3; 
9  +  2V6  +  4(V  +  2V3);  15.25- 5  V0.6. 

13.  Find  the  value  of: 

^^-— ^ — V-T,  given  X  ^  — 7-^ — ~  and  y  ■=  — ,  v        • 
V(^'  +  2/').  given  X  ■=  ^(aV),  y  -=  ^(aV)  ; 

Va  +  ^)-V(l-"),  given  ^  =  -2£L  ; 

V(i+^)  +  V(i-^)  «'  +  *' 


COMPLEX    QUANTITIES.  273 

14.  If  ^(x  +  a  +  h)+-^{x  +  c  +  d) 

=  -y/{x  +  a-c)  +  -^(x-b  +  d),  .\b  +  c  =  0, 

15.  Simplify      Kl+V5)^-2     ^     Kl-V^)^-^     . 


Complex  Quantities. 

Quantities  of  the  form  a-{-b^—l,  in  which  neither  a 
nor  Z>  involves  V~^'  ^^^  called  Oomplex  Quantities.  The 
letter  i  (or  j)  is  frequently  used  as  the  symbol  of  the  diten- 
sive  unit  V~-^'  ^^  ^^^^  a-{-h^—l  would  be  written  a~j~hi. 
So  also  V~^  =  i^x,  ^—x  X  V~  y  =  ^^  VC^y)  ~  ~  V^cV» 
and  i^  =  —  i. 

Ex.  66. 

Simplify  the  following,  writing  i  for  -r^—  1  in  any  result 
in  which  the  latter  occurs : 

1.  v-4;  V-S6;  V-81;  V-8;  V-12;  V-^^; 
^-8;  V-^xV-6;   V-6XV-8;  V-8XV12; 
V-8x-</-8;    V~^XV-20. 

2.  v-^;  V-^';  V-^';  V-^'";  V(-«)'; 
V(-  ^y ;  V(-  ^  ^^') ;  V—  «  X  v^' ;  V-  ^''  x  v-y' ; 

3.  z';    z';    z'-    i' ;    i^ ;    i'' ;    i'^ ;    i^' ■    i'^ ;    z*";    ^*«+l ; 

4.  aixhi]  i-y/x  X  i-y/i/ ;  5^;  ^^-v/^;  i-^—a;  i^—a^; 
i-y/aX  ■\J—a. 


274  COMPLEX    QUANTITIES. 


5.  V-*'; 

V-«';  V-»^  V-*';  -V-i'"; 

V-*'*"- 

6     V-6. 

V-6.      V6    .      Va    .    V-a. 

V-  3 '  V-  3 '  V-  ^  '  V-  ^  ' 

1 

'•     V3    ' 

v-i' 

a             a' 

v(—  «*) .  —  V—  1 .     «'    . 

V-a'    V- 

a''      V-a;    '      V-«    '    ^-«'' 

~c    .    V(- 
^-0^'    V(- 

7.    1-    1- 

1-111         -1         1 

a^i     .       X 

.     -y   .        ci" 

V— a''   *V^'   iy/—y'''       V— ^^ 

8.  v(«-*)x  V(^-«);  V(3»-42/)x  v(4y-3«); 

(3  +  5z)(7  +  4i);    (8-9i)(8- 7z)  ; 
(7-iV5)(7  +  iV10);  (V3-iV6)(V2~«V6); 
(a  +  5i) (c  +  ci)  ;    [a  +  (a  —  l)i] [a  +  (a  +  l)i]  ; 

( V«  +  »V^)  ( V'*  —  *V<^) ;  («  +  ^*)  (<* ""  ^0 ; 

{ai  +  &)  (ai  —  H)  ;    ( V«  +  «V^)  (V*  ~"  * V*)  I 
{ay/b  +  ci  Va;)  (aV^  —  ^i^^) ;    V(l  +  «')  X  V(l  ^ «')  ; 
V(3  +  4 i)  X  V(3  -  40  ;    V(12  +  5?)  X  V(12  -  50  ; 
(1  +  0';  (V«-«V^)';  (5-2iV6)^ 

(a  +  hif  +  (a  -  ^-0' ;    («  +  *«)'  -  (a  -  ^0' ; 

{a  +  Uy  +  (ai-  5)^ ;    [V(4  +  30  +  V(4  -30]^ 

[v(3-4o-v(3+40]^  [v(i+o+va-^)]^ 

(l+if-    (1  +  0^    {a  +  Uy-   {a  +  hif  +  {a~bif; 
(a  +  biy-ia-bif; 

n  +  iysY,  /-i  +  ivsy.  /-i-ivsy. 


COMPLEX    QUANTITIES.  275 

(x  +  ii/y  +  (x  —  iT/Y  ;    (x  +  iyY  —  {ix  +  y)' ; 
(1  +  i^by  +  (1  -  i^by  ;    (a  +  ^^7  +  (a  -  5i)^ ; 


2     y    V      2 

for  all  positive  integral  values  of  n, 
g  4        .         64       .  21        .  5  . 


l  +  zV3'    l-i-yjl'    4  +  3zV6'    V^  +  ^V^' 


7-2iV5  '    l  +  iV^'    l-^'     1  +  ^'     1-^'    (1  +  ^7' 

1  4-^^ .    x  +  yi  ^    a  +  z ya;  ^    ?' y^g  +  y—  5  ^    a~-_bi  _ 
1  —  z  '    X  —  yi'    a  —  z\/:?; '    -y/—  a  —  i  ^h  '    ai-\-h^ 

a  +  i V(l  —  x")  .    V(:y-?/)— V(y  — a:).        1       |       1     . 
l+i,   1-^'.  1,1.1  1       . 


1  +  r    (l  +  ^/      (l-z/'    (l  +  ^r      (l-^y' 

a:  +  y^  I  x  —  yi.  a:+yt      a;— yi     y'a;  +  » -y/y      -^y + ^'Va: . 
a  +  Ji      a  — hi'  a-\-bi      a  —  bi'  ^x  —  iy/y      -^y—i^x' 

^/{\  +  a)  +  »V(1  -  a)      V(l  -  «)  +  t  V(l  +  a) 
V(l  +  a)  -  iV(l  -  «)      V(l  -  «)  -  «V(1  +  «)" 

10.    V(3  +  4i)  +  V(3-4i);    V(3  +  4i)-V(3-4i); 

V(4  +  3«)  ±  V(4  -  3i) ;    VCl  +  2iV6)  ±  VCl  -  2^■V6)  ) 

V(5  +  2  JV6)  ±  V(5  -  2«V6)  ; 


276  COMPLEX    QUANTITIES. 

V(2Vl5  +  30^)±V(2Vl^-30^); 

V[^  +  V(^'  -  «')]  ±  V[«  -  ^V(^'  -  ^')] ; 

11.  Prove  that  both  \{-\-\-i-^?>)  and  i(- 1  -  zV3) 
satisfy  the  equation =  0  ; 

that  (:?;  + (oy + co'^2;7  ::=:r^+y^+ 2;^+ 3  (:r+ toy)  (2/+ 0)2;)  (2; +a):r) 
and  that  (a;+y+2;)(a;+a)3/+w^2;)(a;+(o^3/+(o2;) =^^^+3/^+2;^— 3r?:y2;, 
in  which  w  represents  either  of  the  preceding  complex 
quantities. 

Hence,  prove  that : 
(i.)  \_^a-h~c-\-{b~6) iV37 

=  \2h-G-a\{c-a)  %  V3]'-:  [2c-a-5+  (a-^>)  i^ZJ ; 

(ii.)  16^  +  "^^  +  '^^^  ~  3  uvw 

-:  (a'  -\-h''-\-&~  Zabc)  {x''  +  y'  +  2'  -  3:2:^0), 

\i  u^=  ax  -\-  hy  -\-  cz,  v  =  ai/  -{-  bz  -{-  ex,  lo ~az-\-hx-\-  cy, 

or  if  u=^ax  -\-cy  -\-  bz,  v  =  cx-{-by-{- az,  w  =  bx-\- ay  +  cz. 

12.  Prove  that  i  [  V^  +  ^  +  ^V(10  "  2 V^)]  satisfies  the 
equation  — ^t_  =  0. 

Writing  o)  for  the  preceding  complex  quantity,  prove  that 
(7  +  0)  +  o)^  +  3a>^)(7  ~io'-  oy'-Sio')  -  71,  and 
(^  +  y  +  2;)  (a;  +  cD^y  —  iiih)  (x  —  (i)^y—ioz)  (x—wz  +  u)*z) 
X  (x+(ti'^y+o^'z)  =  x^+y^  +  z^—5x^yz  +  5:ryV. 

Prove  that  [Aa + (b  -  c)  (-y/ b - 1) +  (b  +  c)i^ (10+2^ 5)  f 

=  K«+^)[-i  +  ^V(V5  +  2)] 

+  («- ^)[V5  +  MV5  -  2)]  X  </5-  4c'p. 


SURD    EQUATIONS.  277 


13.    Solve  the  following  equations,  and  prove  that,  if  ?• 
be  a  root  of  any  one  of  them,  the  other  roots  of  that  equa- 


(vi.)  ^  =  0. 

(vii.)  ^  =  0. 

X  -{-  1 

(viii.)  ^'+^-c^ 


(\.)x-  +  l^ 

=  0. 

(ii.)  x*+l- 

=  0. 

(iii.)    3^+1-- 

=  0. 

(IV.)    r  -- 

=  0. 

X  ~l  "        '  x''  +  l 

(v.)  ^^^Zll^o.  (ix.)  (^^^-l)(^-l)^o. 

^    ^  x-l  ^     ^  (a;^-l)(:r^-l) 

§  51.   Surd  Equations. 
Examples. 
Solve  2  +  ^{4:x'  -  9x  +  S)-2x  =  0. 
Here  there  is  but  one  surd,  and  it  is  convenient  to  make 
that  surd  one  side  of  the  equation,  and  transpose  all 
the  rational  terms  to  the  other ;  this  gives 
-y/(4:x'-~9x  +  S)=^2x-2; 
Squaring  both  sides, 

4:x'  —  9x+S  =  4lx''-8x  +  4:.     .-.  X  =  4:. 

Solve  V(4^'  +  19)  +  V(4^^'  -  19)  =  V^^  +  3. 
We  have  the  identity 

(4:x'  +  19)  -  (4a;^  -  19)  =  38-47-9. 
Now  dividing  the  members  of  this  identity  by  those  of 

the  given  equation,  we  have 

■y/(4:x'  +  19)  -  V(4^'  - 19)  =  V47  -  3. 
Adding  this  to  the  given  equation,  then 

2V(4:r^  +  19)-:2V47. 
.-.  4:x^  +  19  =  47,  and  x  -^  ±  -y/l. 


278  SUED   EQUATIONS. 


4 


^x'-l      a4-b 


?>  —  x^       a  —  h 

"  ^-x'       {a-hf 

2 ^ ^{a  +  hy  +  {a-hY  ^  o?  +  ah  +  b' 
"^       {a  +  by  +  ^{a~by      a'-ab  +  b'' 

m  V(l  +  x)-n  V(l  -  a;)  =  VK  +  ^')-  (1) 

Square  both  members  and  reduce, 

.-.  (m^  -  n")  X  -  2mnV(l  -  ^)  =  0.  (2) 

Transfer  the  radical  term  and  square  both  members, 
.-.  {pv?  -  nyx''  -  4mW(l  -  x").  (3) 

.•.(m2  +  n7:i;'=:4mV.  (4) 

±  2??in 


TTh^  +  n^ 


(5) 


The  above  follows  the  usual  mode  of  solving  equations 
involving  radicals;  viz.,  make  a  radical  term  the  right- 
hand  member;  gathering  all  the  other  terms  into  the  left- 
hand  member,  square  each  member;  repeat,  if  necessary, 
until  all  radicals  are  rationalized.  This  method  is  con- 
venient, but  it  does  not  explain  the  difficulty  that  only  one 
of  the  values  of  x  in  (4)  satisfies  (1);  viz., 

-f  2mn 

The  other  value,  ,  satisfies  the  equation 

m^  +  TV 


TYl 


V(l  +x)^ny/0--x)=  ^{m'  +  n'). 


SUED    EQUATIONS.  279 


The  explanation  is  simple.  Squaring  both  members  of 
(1)  IS  really  equivalent  to  substituting  for  (1)  the  conjoint 
equation, 

[m V(l  +  x)  -  nV(l  ~x)-  VK  +  ^')] 

[m^(l  +  x)  +  n^(l-x)--y/{m'  +  n')]  =  0,     (6) 

which  reduces  to  (2)  above. 

Treating  (6)  or  (2)  by  transferring  and  squaring  is  equiva- 
lent to  substituting  for  it  the  equation 

[m  V(l  +  x)~  n V(l  -  ^)  -  V(^'  +  ^')] 
X  [mV(l  +  ^)  -  n^(l  -x)  +  -y/im'  +  n')] 
X  [m V(l  +  x)  +  ny^(l-x)~  V(^'  +  n')] 
X  [m  V(l  +  x)  +  ^  V(l  -  ^)  +  V('^^'  +  ^')]  =  0,  (7) 
which  reduces  to 

[(m^  —  92^)  :u  —  2mn^(l  —  x"^)] 

[(m''-n')x  +  2mnV(l  -  ^')]  ^  0,  (8) 

which  further  reduces  to  (3). 

Thus  the  whole  process  of  solving  (1)  is  equivalent  to 
reducing  it  to  an  equation  of  the  type  A  =  0  and  then  mul- 
tiplying the  member  A  by  rationalizing  factors.  Thus, 
instead  of  solving  (1)  we  really  solve  (7),  that  is,  a  conjoint 
equation  equivalent  to  four  disjunctive  equations.  (See  page 
191,  §  42.)  Now  the  values  given  in  (4)  will  satisfy  (7), 
the  positive  value  making  the  first  factor  vanish,  the  nega- 
tive value  making  the  third  factor  vanish,  while  no  values 
can  be  found  that  will  make  either  the  second  or  the  fourth 
factor  vanish. 

Hence,  if  one  of  such  a  set  of  disjunctive  equations  is 
proposed  for  solution,  the  conjoint  equation  must  be  solved; 
and  if  there  be  a  value  of  x  which  satisfies  the  particular 
equation  proposed,  that  value  must  be  retained  and  the 
others  rejected. 


280  SURD    EQUATIONS. 


This  process  is  the  opposite  to  that  given  in  §§  42  and  43  ; 
there  a  conjoint  equation  is  solved  by  resolving  it  into  its 
equivalent  disjunctive  equations.  The  two  processes  are 
related  somewhat  as  involution  and  evolution  are. 

Further,  it  should  be  noticed  that  just  as  there  are  four 
factors  in  (7)  while  there  are  only  two  values  in  (4),  it  will 
in  general  be  possible  to  form  more  disjunctive  equations 
than  there  are  values  of  x  that  satisfy  the  conjoint  equation, 
and  consequently  it  will  be  possible  to  select  disjunctive 
equations  that  are  not  satisfied  by  any  value  of  x,  or,  in 
other  words,  whose  solution  is  impossible. 

This  will  perhaps  be  better  understood  by  considering 
the  following  problem  : 

Find  a  number  such  that,  if  it  be  increased  by  4  and  also 
diminished  by  4,  the  difference  of  the  square  roots  of  the 
results  shall  be  4. 

Keduced  to  an  equation,  this  is 

V(^  +  4)-V(^-4)-4.  (9) 

Kationalizing,  this  becomes 

[4-V(^  +  4)  +  V(^-4)] 
x[4-V(^  +  4)-V(^-4)] 
x[4  +  V(a^  +  4)  +  VG-e-4)] 
x[4+V(«+4)-V(-^-4)]=0,  (10) 

whicli  reduces  to 

[24  -^  8  V(a^  +  4)]  [24  +  8  V(^  +  4)]  =  0  ; 
that  is,  9  —  (a;  +  4)  —  0,  or  a;  =  5. 

Now  x^b  satisfies  (10)  because  it  makes  the  factor 

4-V(^  +  4)-V(a--4) 

vanish,  and  it  is  the  only  finite  value  of  x  that  does  satisfy 
(10),  or,  in  other  words,  there  are  no  values  of  x  which  will 
make  any  of  the  factors 


SURD    EQUATIONS.  281 


4- V0^  +  4)t- VC^-4), 

4  +  V(^  +  4)  +  V(^'-4), 

or  4+V(^  +  4)-V(^'-4), 

vanish.     There  is,  therefore,  no  number  that  will  satisfy  the 
conditions  of  the  problem. 

It  will  be  found  that  as  x  increases  -\/{x  +  4)  —  ->/(:?;  —  4j 
decreases ;  hence,  as  4  is  the  least  value  that  can  be  given 
to  X  without  involving  the  square  root  of  a  negative,  the 
greatest  real  value  of  ^{x  +  4)  —  -yjix  -  4)  is  ^8,  which 
is  less  than  4.  We  see  by  this  that  our  method  of  solution 
fails  for  (9)  simply  because  (9)  is  impossible. 

5.    ^[{a  +  x)(b  +  x)]-yj[{a-x){b-x)] 

-  V[(^  -^)(J>  +  ^)]  -  V[(«  +  ^)  (^  -  ^)].  (1) 

Collecting  the  terms  involving  -y/^a  +  x)  and  -y/(a  —  x), 
respectively  the  equation  becomes 

[^{a+x)-^{a-x)]  [  y/{h-\~x)+^{b  -  x)]  -  0.       (2) 

This  is  satisfied  if  either 

-yjia  +  x)~.  V(a  ~  ^)  =  0,  (3) 

or  V(^  +  ^)  +  -y/ip  -  ^)  =  0.  (4) 

The  rational  form  of  (3)  is  {a  +  x)  —  (a  —  x)  —  0,  which 
is  satisfied  by  a;  =  0,  and  this  also  satisfies  (3). 

The  rational  form  of  (4)  is  {b  -\-  x)  —  (b  —  x)  =  0,  which 
requires  x  =  0]  but  this  does  not  satisfy  (4).  Hence, 
the  second  factor  of  the  left-hand  member  of  (2)  can- 
not vanish. 

Therefore,  the  only  solution  of  (2),  and  hence  of  (1),  is 
x=^0,  derived  from  (3). 


282  SURD    EQUATIONS. 


6.    ^{^a  +  x)+^{a-x)  =  ^{2a). 

Cube  by  the  formula, 

{u  +  vf  =  u^  +  v^  +  '^uv{u-^  v). 

.-.  {a  +  x)  +  (a-  x)  +  3  {/[2a(a'  ~  x')]  =  2a. 

.\2a(a''-x')  =  0. 

,\  x  =  it  a. 

Both  these  values  belong  to  the  proposed  equation. 

The  rationalizing  factors  of 

-^(a  +  x)  +  -^(a-x)-^(2a)  =  0 

are      -{/(a  +  x)  +  o)  ^{a  -  x)  -  co^  -^(2  a), 

and     -Y/(a  +  ^)  +  w^-\/(<^  — ^)  —  «-\/(2a). 

See  Exam.  11,  page  276. 

The  remarks  on  Exam.  4  will  apply  mutatis  mutandis 
to  equations  of  this  type. 

7     a/(^  +  xy  +  ^{a'  -  x')  +  ^{a  -  xj  _  ,^' 

•    ^{a  +  xj  -  ^(a^  -  x')  +  ^{a  -  xf      '''  ^  ^ 

Assume  -^{a  {-x^  =u  and  ^(a  —x)  =  v. 

j.u^-\-v^  =  2a  2ind  u^ -v^  =  2x. 


u   -  V  _x 
'  '  u^  +  v^      a 

Also  (1)  becomes 

V?  -\-uv-\-  v^ 


u  —  v 


(2) 


(3) 


Multiply  both  numbers  by 

u^  -  v^         u~v 

•  • ^=  c • 

u^  ~\-v^         u-{-v 


u-{-v 


SURD    EQUATIONS.  283 


.-.  by  (2), 

f  =  ,?i^".  -(4) 

a        u-\-v 

Again,  adding  and  subtracting  denominators  and  numer- 
ators in  (3), 

uv         c—1 
Adding  and  subtracting  2  (denominators)  and  numera- 
tors in  this, 

V?  —  2uv -\- v^  _  3  — c 


(u  —  v 


.'.  substituting  by  (4), 


—  =  ^^ 


dc-1 

8.    [^(x  +  a)  +  ^X^-a)J[^{x+a)--^(x-a)]  =  2c.  (1) 

Assume  u  =  -y/^x  -\-  a)  and  v  —  -^/(x  —  a) , 
and  (1)  becomes 

(u  +  vy(7i  ~  v)  ^  2c, 
or       (u  +  v)\u'-v'')  =  2c.  (2) 

Also       ii^—v^  =  2a 

or       (u'  +  v^)(u'-v')  =  2a,  (3) 

and    u^  +  v^  =  2x.  (4) 

From  (2)  and  (3), 

(u  -  v)\u'  -v')  =  4:a~2c.  (5) 


284  SURD   EQUATIONS. 


.-.  (2)  X  (5), 

(u'  -  vj(u'  -  vj 
or       (u'  -  vj  =  ^c(2a-c),  (6) 

Also(3y  +  (6), 

[(u'  +  vj  +  (u'  -  vj]  {v?  -  vy 

=^^(o?  +  2ac-c^) 

or       {u'  +  v') {u" -  vy  -=-2{a'  +  2ac-  c"). 
Substituting  by  (4)  and  (6), 

2x  -yj{2ac  -  c')  =  0"  +  2ac  -  c\ 

9.    [^(a+x)+  ^(a-x)J[^{a+x)+-^(a-x)]  ^2cx. 

Divide  the  terms  of  the  identity 

-^(a  +  xy  -  -^(a  -  xy  =  2x 
by  the  corresponding  terms  of  the  equation. 

•         4l/^'  +  ^\  __  g  +   1 

"\\a-x)      c-i 

,  a  +  x __fc+iy 
a  —  x      \c—\J 

10.    ^{a  -  xf  +  -^[(a  -3-)(b-  x)-]  +  ^{h  -  xy 
^-^{a'  +  ab  +  b''). 

Divide  the  terms  of  the  identity 

i/{a  -  xf  -  -i/{b  +  xy  =  a-b 
by  the  corresponding  terms  of  the  equation. 

a  —  b 


.■.-^{a-x)-^{b-x)- 


^{a^+ab  +  b'') 


SURD    EQUATIONS.  285 


Cube,  using  the  form 

{u  —  vy  =  u^  —  v^  —  Suv(u  —  v). 
{a  —  x)  —  (h  —  x) 

-^^[{^OL~x)(h-x)]  ""-^ 


a'  +  ab  +  h''      ^  o}  +  ah  +  b' 

ah 


.'.  (a  —  x){h  —  x) 


^{o}  +  ah  +  by 
a'b' 


(p}  +  ab  +  by 
a  form  solved  in  (C),  page  231. 

11.  (g  -  ^y  v(a  -  x)-\-{x-  br^{x  -  b)  _  ^^  ^ 

{a  —  x)^{a  ~x)-\-{x-  b)-y/{x  —  b) 

Write  a~-b  in  the  form  {a  —  x)  -\-  {x  —  b),  and  multi- 
ply by  the  denominator  of  the  left-hand  member. 

.-.  (a  -  xy-^{a  -x)  +  ix-  bf^Xx  -  b) 
=  (a-xy^(a-x)+(x  -  by-^(x  -  b) 
+  {a  —  x){x  —  b)[^{a  —  x)  +  V(^~*^)]- 

.-.  {a  -  x) {x  -  b)[^{a  -  x)  +  V(^'  -  ^)]  =  0 

,'.{a  —  x)  =^0,    or  X  ~  b  —  0, 
or     ■y/(a-x)  +  ^(x  ~b)^0. 

Xi  —  a,   x^i  —  b. 

The  equation  -^/{a  —  x) -\-  ^{x  -  b)  =  0  has  no  solu- 
tion, for  the  sum  of  two  positive  square  roots  cannot 
vanish. 

The  solution  x  ^=  i  (a-}-  h)  belongs  to  the  equation 

V(a  -  ^0  -  -  V(^  -  ^)  ^  0. 


286 


SURD    EQUATIONS. 


12. 


b-\-x  _ 


Square   both   members,   subtract  4,   and    extract  the 
square  root. 

3-\-x  _ 


:±V(^^-4). 


a  —07 


=-  i  [c  ±  V(^'  ~  4)]  =  e,  say. 


'Z)  +  o; 


(^-^). 


a  +  5 


1  +  ^^ 


,\x=  t 


(a 


-^)+(«  +  5)l^] 


c'-Sc-2 


Or  thus,  cube  both  members. 

^4"^  a  —  a; 

.  (a  -  xY  +  {h  +  xy 
(a  —  x)(b  -{-  x) 

.  V{h  +  x)-{a-x) 
"[(h  +  x)  +  {a--x) 

a  +  h  6-1^0  +  2 

Prove  that 


c^-3c  +  2 


l-e" _c+l    \c-2 
l  +  e"     c- 


l    fg-2 
We +  2 


if      2e  =  c?iV(^'-4). 


SURD    EQUATIONS.  287 


Ex.  67. 

1.  ^(x  +  4:)  +  ^X^-S)^7, 

2.  V(3^+l)  +  V(4^^'  +  4)-l. 

4.  -y/^inx)  —  -yj(nx)  —  7)i  —  n. 

5.  -yj{hx)  +  -^(ab  +  hx)  =  ^x. 

6.  V^  +  V(^+3)  ^ 


7.  ^{ax  +  x')  =  (l  +  x). 

8.  ^(17:r~26)-| 

\x 
10.  6  +  :r— VC^'  +  ^O^^'- 
12.  V(2ri;-27a)--9V^- V(2^)- 

13  5V(2.2;-l)  +  2V(3.T-3)__^,, 

*  4V(2a;-l)-2V(3^-3)        '^' 

14  V2a;+V(3-2r^)_3 
V2-^- V(3-2a;)      2 

15  2^(3a;  +  3)  +  -^(7:.  +  8)_g 

•  2^(3a;  +  3)-^(7:i;  +  8) 

16.  33  [13  -  2  V(^  -  S)]  ==  3  [13  +  2 V(^  -  5)]. 

18.  v(^ + g)  +  v^  ^  v^  +  yg 


V(a;  +  c)  -  V^       V^  ~  V 


'a 


288  SURD    EQUATIONS. 


19     V^-  +  28_  v:r  +  38 

V^  +  4  ~'  v^  +  ^* 

^2:i'  +  9       -^2:^+15 

21  V^W-2a_  Va;  +  4a 

■y/X+b  ^x+3b 

22  3^-1    __1  +  V3r^ 
V3^  +  l  2 

23.  V^--V(^-^)^^. 
Va  -f  ■  V(<^  —  ^0 

24.  V^^+ V^-^ 


25.    ^><-'  +  1  +  V(q^'^'  -  1)  _  ^^ 


a^+  1  —  ^(aV  -  1) 


26.    l-Vri--</(l-^)1^a. 

i+V[i-V(i-^)] 

27     ^(a;+l)-^(^-l)_l 
^(x-l)+^(a;-l)      2 

28.  .^(l-^)+^(l  +  a;)  =  ^3. 

29.  ^(3  +  »)  +  ^'(3-^)  =  ^7. 

30.  ■^(a;+l)-^(a;-l)  =  ^ll. 

31.  ^(a  +  a;)  +  ^'(a  -  ^)  =  a/^- 

32.  ^(1  +  V^)  +  ^(1  -  V^)  =  2. 

33.  y/x  -  V[«  —  V(«^  +  ^)]  =  i  V«- 

34.  7(25  +  a;)  +  ^(25  -  a;)  =  2. 

35.  -+V(«^  +  -0-^^y 


SURD    EQUATIONS.  289 


36.  v(i  +  ^)  +  V[l  +  ^  +  v(i  -  ^0]  ==  V(l  -  ^)- 

38.    V(l+^^  +  ^^')+ ■^/(l"^  +  ^'')="^^^^• 
39.   v(^'  -  ^')  +  ^ V(^' "  1)  ^  ^'  (1  -  ^')- 


40.  ^^  "  ^'       -:    V(^^)  +  ^  g. 

y/'(bx)  -\-c  n 

41.  V(2^''  +  5)  + V(2^^-5)-Vl5+ V^- 

42.  V(S^'+  10)+  V(3^'-  10)  -  Vl^  +  V37. 

43.  V(3  ^''  +  9)  -  V(3  ^'  -  9)  -  V34  +  4. 

44.  V(S^-^^  +  ^')+  V(2«-2^>  +  :i'')-- V^+  V^' 

45.  VC^'^'  -  3<^'  -  2x'')  +  V(3a'  -  3^>2  -  x'')'-^a  +  x. 

46.  [  V(a  +  a;)  +  V(«  -  ^)]  [  V«  +  V(«'  +  ^')]  ^  2a;. 

47.  Vi^'  +  ^-^)+^(^'-^^^)--^jJ^' 

"■  >l(^)-^(-$-;-^)-v(.-»-). 

49.  V[(2«  +  ^y  +  b']  +  V[(2a  -  a-y  +  6'']  =  2a. 

50.  rV(«-^)+V(:r-&)T^_ 

V(«  —  x)  ~  V(^  ~  ^) 


51. 


52. 


53. 


54. 


V(i+^^)-V(i-^'0    * 

^(l  +  x'^-^a-x')      b 

^(l  +  x')+-^(l^x')_a 
^{l  +  x')--^(l-x?)      h 


290  SURD    EQUATIONS. 


^{l  +  x')--^{x'-l)      b 

V(^''^  +  l)-V(^^^-l)      ^ 

57.  V(4^  +  ^  "  ^^)  -  2  V(^  +  b~-2x)  =  -^h. 

58.  V(3^-2^  +  2ri;)-V(3«-2^-2:r)-^2va. 

59.  V(2«  -  ^  +  2a:)  -  V(10a  -  9 ^  -  6:?;)  -=  4  V(«  -  ^)- 

60.  V(3^  -^b  +  bx)  +  VG^  -  a)  -:  2  V(^  +  a). 

61.  V(^^  -  4^  +  bx)  +  V(^'  -  a)  -=  2  V(2^  —  2b). 

62.  V(5^  -  ^<^  +  4^)  +  V(^^  —  3a  —  4^))-=2  V(^'  +  a). 

63.  V(2^*+H2^)+ V(10«+9'^-6^)  ^  2  V(2a+^-2a;). 

64.  2 V(2a+H2:r)+ V(10a+5-6:^)  =  V(10^+9^-^^)- 

65.  V(2a-13^+14a:)+ V[3(^-2a+2:r)]  =-  2V(2a-5+2:^). 

66.  ^[^{1  a  +  b  +  x)]-  ^\a+1  b - x)  ^  2-y/{1  a  +  b  - x). 

67.  v[(^^  +  ^)(^  +  ^)]  +  V[(«  -  ^0(^  -  ^)]  -=  2  V(«^). 

68.  V[(«  +  ^)(^*  +  ^)J  -  V[(^  -  ^)(^  -  ^)]  -=  2  V(^^). 

69.  -^/(aa;  +  x^)  —  y/{ax  —  o;^)  —  ^{2  ax  —  a^). 

70.  ^{ax  -  ri;')  +  y  («^  +  ^')  "=  V(2 «^'  +  «')• 

^2  x-^^{ax)      a  +  -^J{ax)  __x  —  a 
a  —  ^{ax)      X  —  -y/iax)  a 

73  Vr {a  +  :^0  (^  +  ^)1  +  V[(a  -  ^)  (^  -  ^)1 .__     \a 

'  ^\(a  +  x)(x  +  b)]-^[{a-x){x~b)]      \b 


74. 


V[(a  +  ^0  {x  +  ^^)]  -  V[(«  -  ^)  (:^  -  ^')]       \^ 

|3a- 
\3a- 


j3a  -  2^>  +  2a:  _.,  [  Va  +  V(2a  -  2^>)f 
26-2a:  2^)-a 


SUED    EQUATIONS.  291 


75.  ^(a  +  x)  +  -^(a-x)  =  2-^a. 

76.  i/(a  +  xf  -  ^{a'  -  x')  +  ^(a  -  xf  =  -^a'. 
^(1  +  xy  +  ^(1  -  X')  +  ^(1  -  xY  _  .„ 

78.  ^(i+xy  +  ^(i-  xy  =  2 J  ^(1  -  x'). 

79.  ^(3  + a;) +  ^(3 -a;)  =^6. 

80.  ^(1  +  xy  +  ^(1  -xy  =  5  [-{/(I  +  x)  +  ^(1  -  x)]\ 

81.  s/(14  +  a:)^  -  -{/'(ISe  -  :^-)  +  a/(14  -  ^y  =  7. 

82.  [^(9  +  x)+-^{9--x)]-^{8l-x')  =  12. 

83.  [^(14+a;y-^(14-a;)^][^(14+a;)— ^(14-^a;)]  =  16. 

84.  [^(57+x)'+^'(57-a;)^][^(57-a;)+^(57+a;)]  =  100, 

85.  5[</(41+z)+^(41-a;)P  =  8[ V(41+^)+V(41-a^)]. 

86.  [^(:r  +  5)  +  ^(^-5)]'[^(a;  +  5)  -^(a:-  5)]  =  2. 

87.  [^(a;+l)+^(a;-l)][V(^+l)  +  V(^-l)] 

=  26[^(a;+l)-^(^-l)]. 

89.  2[^(l+^)^+^(l-a;')]  =  (c'+l)[-^{l+x)+-^'a~^)J- 

90.  ^(a  +  a;)  +  ^(a  -  a;)  =  ^c. 

91.  [^(a  +  a;)  +  ^(a-a;)]^(a^-a;^)  =  c. 

92.  .^(a  +  xy  =  ^(a''  -  x')  +  ^(a  -  a;)''  =  ^c\ 

93.  [^(a  +  xy  -  ^(a  -  :»)^]  [^(a  +  x)  -  -^(a  -  x)]  =  c. 

94.  [^(a  +  xy  +  ^(a - xy][-^{a  +  x)  +  ^{a - x)]  =  c. 

95.  (a  +  a;)^(a-a;)-(a-a-)^(a  +  a;) 


292  SURD    EQUATIONS. 


96.  (rt  +  x)  -^{a  +  x)  —  (a  —  x)  -^(a  —  x) 

-^c[^(a  +  x)---^(a-x)l 

97 .  [i/(a  +  xy  -  -^(a'  -  x')  +  -{/(a  -  :r)^]^ 

=  4^(a  +  r^)  +  -^(a-a;)]. 

98.  [^(a+x)+^(a-x)J  =  (c+l)[-^(a+x)+^(a-x)l 

99.  [^(x  +  a)-  ^(x  -  a)]  [  V(^  +  a)  +  V(^  ~  «)]' 

-4^(^  +  a)  +  ^(a;-a)]. 

100.    V(^  -  Q^)  +  V(^  -'b)_     \x  -  g 


V(^  —  a)  —  V(^  "  ^) 


\x 


^{x-a)  +  -^{x-h)  _     la-x 


b  +  x_^ 
a  —  X 


102.    ^^-J.^-  =  c. 


103.  f^~^  I       p~^ 

104.  3|«_z^-_    31^+^ 


'»H(f^3"  ""' 


\  \a  —  xj 

106.  J^^+_4^±^  =  c,     109.     .4^-J^ 

107.  J^^-J^Zli:-  =  c.     110.    .fc+.te=c. 
\b  —  x       \a  —  x  \6  +  ^      \a— a; 


■X 


108.      ^^^-^    5|i±£  =  ,.      111.      e|"_    6^==,. 
\6  +  a;       \a  — a;  \6  — a;      \a— a; 


112.  v(«--^y+v(^-^y-c. 
V(«  — «)  +  v(* "~  ^) 

113     V(«-«y  +  V(&-.rf  _^ 

•  [v(«-a:)+v(6-^)r    ■ 


SURD    EQUATIONS.  293 


114.  ^(^'-^r+^(^-^f-,, 

-yj{a  —  x)  —  ^{b  —  x) 

115.  rv(a-^)  +  v(^-^)?_,^ 

■yjia  —  x)  —  V(^  —  ^) 

116.  v(a-^y+v(^-^y^^^ 

117     V(a-^y- V(^-^)'^^ 
•    [^{a-x)~^{x~h)J 


119. 


V(a  -  ^)^  +  V(^  +  hf  __  {a  +  by 

V(a  ~x)  +  V(^  +  ^)       4  v[(«  -  ^)  (^  +  ^)]' 


:?;' 


(g  —  x'^)  ^{a  —  r^:'^)  _ 


I 


120.       '  ^  —  ;  ^\,   ^  =  g. 

122.    -^(a  -  :r)^  -  -{/[(a  -x){x-  h)]  +  -^(o;  +  by 
=  ^{d'  -ab  +  b'). 

■^(a  —  x)  +  -^(x  —  ^) 

124.    Q^  -V(a  --x)  +  b  ^(x  -  ^>)  _^  ^^ 

V(a  —  :^)  +  V(-^  —  ^) 

125  V(^  -  Q^)  +  Vfa  +  (i)  —  V(2 g)  _   4/^  +  g 

126  VC^  —  ^)  +  V^  —    ij^LH^, 
■yy(x~b)+-y/c       \a;-6* 

127.    ^(g  -  xy  -  ^[(g  -  :r)  (a;  +  b)]  +  -{/(a;  +  &)^ 
=  -^(d'-ab  +  b'). 


294  SURD    EQUATIONS. 


128.  \  {/(a  --  xY  ~  ^[(a  -  x)  (x   -  ^)1  +  ^(x  --  bJl' 

129.  [i/(a-xy+^(b^xyf  -  (a+b)\{/(a x)+^(b+x)l 

130.  ^(a  ~  x)  +  -^(x   -  b)  -  ^c. 

131.  ^(a  +  xf  -  ^(a  -  xf  =  ^{2 ex). 

132.  ^(a  -  ^0'  +  </Kci  ~  x)  (b    -  X)]  +  -{/(b  -  -  xf  -  ^^l 

133.  ^(a--.^0'~-v/[(«---^0(^^^)]-f  v/(^'  +  ^)7 

=  c[{/(a-x)  +  -^(x  +  b)l 

134.  [-^(a  -  ^0  i-  </'(^  +  ^)]</{('^  ~  ^^"^  (^  +  ^)]  ^  ^• 

135.  ^(a  -  o;)'^  +  i/(x  -  ^)'^  -  c  [-^(a  -  :^0  +  -^'(^'  -  ^)]^ 

136.  :r+^(a^-rrO=-— -4 - 

137         ^'     _x+-^(2b'~x') 
x'~b'      x-^(2b'-x') 

138.  (a  +  a;)^(a  +  :?;)  +  (a~r?;)^Ya-^) 

=  a[^(a  +  x)+^(a-x)l 

139.  (a  +  .r) -^(a  -  ^)  +  (a  -  :r)  ^(a  +  a;) 

-  a[-^(a  +  ^)  +  '^(a  -  x)]. 

140.  ^(26  --x)+  ^(x  -  10)  -=  2. 

141.  [^(a-a,0  +  -^Cr--Z^)7-4V(«--:i-)+V(^^'-^)]- 

142.  (a  —  x)-^(a-x)  +  (x  —  b)^(x  —  b) 

^(a-b)[^(a-x)  +  ^(x-b)l 

143.  [</(«  -  x)  +  -^(x~b)J[-y/(a  -x)+  V(^-  6)] 

=  c(a  +  ^  -2.r). 

144.  [^'(a  -  :r)  +  -^(b  ~-  x)]  [^(a  -x)  +  ^(b  -  -  x)Y 

-=c[^(a-x)---:^(b  -x)l 

145 .  a  V(l  +  ^')  -  ^  V(^'  +  <^')  "=  ^- 


SURD    EQUATIONS.  295 


146.  (a-x)-^(x  —  b)-\-(x  —  b)-{^(a-x) 

147.  [i/(a-x)  +  ^(b  +  x)J  =  c[^la-xy+^(b  +  xyi 

148.  [-^(a  -x)  +  -^(b  +  x)J  =  c  ^[(a  -x)(b  +  x)], 

149.  ^(a  -  xj  -  yib  -  xj  -  c^{a  +  5  -  2^0- 

150.  Aj{a-x)-\--^{x~b)=^^c. 

151 .  y{a  -  ^0  +  -\/(^  -  ^)  ==  a/^- 
(a  -  a;)  ^'{a  —  x) -\-  {x  —  b)  -^{x  ~b)  _ 


152. 


(a  -  x)  ^{x  -  b)  +  {x  +  b)  ^{a  -  x)  ' 


153.    (<^  -  ^)  -v/(^  —  ^)  +  (fi  —  x)^{a  —  x)  _^ 


I 

"     154. 


-^{a-x)  +  ^{x-h)_  c 


155 


■^{a-x)  —  -{/(x-b)     a  +  b-2x 

[^(a-x)  +  ^(b~x)J_ 
^ia-x)~-^{b-x)        ''■ 

156.  {a  —  x)^{a  —  x)-{x-b)-y{x  —  b) 

==c[-^{a-x)--^{x~b)]. 

157.  {a-x)-{/{x  +  b)-(x  +  b)-^{x-a) 

=  c[-^ia-x)~^{x  +  b)]. 

158.  [^(a  -  xY  +  ^(a;  -  6)'']^[(a -x){x~  b)]  =  ^. 

159.  [^■(a^x)--^{x-b)Ji^{a-xy-^{x-by-]  =  c. 

160.  [^(a_a;)^-^(2;-6y][V(a-«)  +  ^(A--^')J  =  f- 

161.  [^(a  -  xf+  -^{x  +  byj  =  c[^(a  -  x)  +  </{x  +  b)]. 

162.  V(«'-«'-^')  +  V(«'-^'-c')-V(^'-c'-a')  =  «- 
163     V(a^  +  2:g)  +  V(a'-2a:) 

VK  +  2a;)  -  VK  —  2a;) 
_  w-l-g'      V('7)z'a;  +  2)  +V(mlr-  2) 
a''        V(^''^  +  2)  -  V(^'^  -  2) 


296  SUED   EQUATIONS. 


164.  V(^'  -  «')  +  V(-e'  -  ^'0  +  V(*'  -  c')  =  X. 

165.  [■^{a-x)  +  -i^b  -  x)] [ ■</(«  -  «)  -  -^(b  -  x)]  =  c. 
166  -^(«-a;)-^(a;-^')_a  +  ^'-2a; 

167.  -^{a  +  x)+-^{a-x)=-^{2a). 

168.  r^(«-^)'+-^(^-^)T^^_6. 

Write  u  for  -^/(a  —  a;),  and  v  for  -^(a;  —  b). 

169.  v(5-3a;  +  a;*)  +  V(5-3y  +  2/^)  =  6;  .'C  +  2/  =  3. 

170.  ■^{x  —  xy)  +  ^{y-xy)=a;    x  +  y  =  b. 

171.  ^(a;  +  m)  + V(«/  +  w)=a;   a;  +  2/  =  6. 

172.  ^(143  +  «)-A/(y-18)  =  l;   a;-2/  =  50. 

173.  v(«y)  +  V[(i  -  *)  (1  --  2/)]  =  « ; 

174.  a;y+V[(l-^0(l-yO]  =  «6; 


2a6 

175.  v(«  -  «^y)  +  V(y  -  ^y)  =  « ; 
V(»  -  x'')  +  v'(3/  -  y')  =  ^• 

176.    .rS  +  y§  =  3(a:i-,yi):=3a;. 

■    \y'~h'      d'-x")   ■  \f  +  Jj'^a'  +  x') 
xy  —  ab. 
178.    {x  +  y)i  +  (x-y)i:=ai; 
(x'  +  y'y  +  {x"  -  -  f)^  =  ai 


CHAPTER  IX. 

Cubic  and  Quahtic  Equations. 

§  52.    The  Cubic.    Let  the  general  cubic  equation  be 

ax' -\-Zhx^ -\-Zcx-\-  d=-^.  (1) 

Let  y=^  ax-\-  h.  (2) 

Substitute  -^ for  x  in  (1),  and  multiply  the  resulting 

equation  by  a^, 

f  +  ^{ac  -  P)y  +  (a'd-Sahc  +  2b')  =  0, 
wbicli  may  be  written 

2/^  +  3^y  +  (?  =  0,  (3) 

in  whicb  11=  ac  —  IP',  (4) 

and  G-=a^d-^abc-\-1h\  (5) 

Assume        y  =  o)-\/ri  +%o^  -v/^2,*  (6) 

in  wbicb  w^  =1.  (7) 

Cube  (6),     f  =  n  +  r,  +  3  ^(nr.Xoy  -^n  +  ^'  ^/r^) 
=  n  +  r^  +  S-^(r,r,)y', 
3/^~3^(nrOy-(n  +  r,)-0.  *  (8) 

Equate  coefficients  in  (3)  and  (8), 

ri  +  r,  =  -G,  (9) 

and  -^(nr,)  =  ~ir.     .\r,r,  =  -IP.  (10) 

*  Throughout  this  chapter  the  symbols  -^  and  -J/  will  be  used  to 
denote  the  corresponding  arithmetical  roots  of  the  quantities  they 
operate  on.  General  roots  will  be  denoted  by  exponents.  See  §  50, 
page  264. 


298  CUBIC    AND    QUAHTIC    EQUATIONS. 

(9)   and   (10)  show  that  Ti  and  rg  are  the  roots  of  the 
quadratic 

r''+Gr-IP  =  0,  (11) 

which  may  be  written 

(2r  +  Gf  -  (G'  +  AIT')  =  0.  (12) 

By  (4)  and  (5), 

Q2  j^  4^3  _  a'(a'd'+iac'-6abcd+Wd~Sb'c') 

=  a'A,  say.  (13) 

.\(2r+Gf-a'A^0.  (14) 

.'.r  =  —  iGzt:  ia-y/A. 

Let.      r,  =  -iG-i  a  V^.  (15) 

/.r,  =  -iG+ia-^A.  (16) 

Substitute  these  values  for  rj  and  7*2  in  (6). 

.-.  3/  -  co^(-i(?-iaVA)+cu^-^(-i(?+iaVA) 

=  -io-^(iG+ia-^A)-oy'-^(iG-ia-y//l).      (17) 

Substitute  this  value  of  y  in  (2). 

r.x  =  l-[~h-uy^(iG+i  a  VA) 

-w'-^(iG~ia-y/6.)].  (18) 

Hence,  if 

x,:=^^[-b-^(iG+iay/A)~-^(iG-ia^A)l 
a 

=  -[-^  +  J(l  +  W3)^(i(?+JaVA) 

'  +Ki-^VS)-\/(i^-2«VA)],       (19) 

X,  -  -[-  5  +  J(l  -  i-V^)</ii  ^  +  i«VA) 

+  J (1  +  ^  V3)-\/(i  G^  -  i « VA)] ; 

inwhicha^A=G^^  +  4^^ 

G  =  a'd-3abc  +  2b\  (20) 


x.^ 

a 


CUBIC    AND    QUARTIC    EQUATIONS.  299 


If  the  cubic  have  one  or  more  rational  linear  factors,  the 
above  method  of  solution  should  not  be  attempted ;  but 
such  factor  or  factors  should  be  determined,  and  the  cubic 
resolved  into  its  equivalent  disjunctive  equations,  and  these 
solved.  (See  Note,  page  233.)  If  the  cubic  have  no  rational 
linear  factor,  -y/(i  (9^  +  ia-^/A)  will  not  he  reducible  to 
the  form  u  +  -yjv ;  but,  if  the  cubic  have  such  factor, 
■yJ{iG+  Ja-y/A)  will  be  reducible,  and  the  reduction  may 
he  effected  hy  resolving  the  cubic  into  its  equivalent  dis- 
junctive equations,  and  solving  these. 

Examples. 

1.  Solve  9a^-9x~4:=^0. 

Assume      x  =  o)  -^/n  +  ^^  a/^2- 

/.x^=^ri  +  r,  +  S-^(r,r,)x. 
•  '•n  +  r,  =  ^, 
and         ^(r.r,)  =  ^. 

•"•  "^1  —  i  ^^^  ^2  =  i- 

.••^i=^*  +  ^i-=i(-^9+-^3). 

2.  Solve2:r^  +  6^^+l-=0. 
Mrst  Solution. 

Assume      y  —  x-}-l,  and  substitute  for  x. 

/.2y'-6y+5  =  0. 
Assume      y  =  o)  -^7\  +  w^  ^7*2, 

cube  each  side,  and  compare  coefficients  with  those 

of  the  equation  in  y, 

n  +  ^^2  =  — I; 

r,r.  =  1. 


300  CUBIC   AND   QUAIITIC   EQUATIONS. 


•.r'  +  f7'+l--0. 

.•.n--2, 

and 

r.  =  -h 

*     • 

■■"'=  </'  li 

^y. 

•-.-      1      ^2      ^^. 

Second  Solution. 

Let 

z~  X  ^,  and  substitute  for  x. 

. 

\z^  +  ^z  +  2^0. 

Assume 

Z  =  (i)  -y/n  +  W^  ^r2. 

. 

•.n  +  r,--2; 

--(i^)'-«- 

. 

•.n  =  -4, 

and 

r,  =  2. 

. 

-.21=^2-^4. 

•:r-        1 

*         ^2- -^4 

^4  +  ^8  +  ^16 
2-4 

i'+^'+U^ 


3.    Solve  a;  +  3/'^  =  7; 

^  +  3/=ll. 


.\x+(ll-xy  =  7, 

.\x'-22x''  +  x  +  lU  =  0, 

.-.  (:r  -  3)  (rr^  +  3 o;'^  ~  13y  -  38)  =  0. 


CUBIC   AND    QUARTIC    EQUATIONS.  301 

Therefore,  either 

^-3  =  0, 
or  else    rr' +  3^^  -  13a;  —  38  =  0. 
Let  z  =  X  -}~1, 

and  substitute  for  x  in  the  latter  of  these  disjunctive 
equations. 

.•.2^-162;-23=-0. 
Assume      z  =  (i}  -^Ti  +  a>^  -^r^. 
.•.n  +  r2-23, 
and         ^/(nrs)  =  -i/. 

.•.r==i[23±V(23^-^W^)] 
=  -i-(23±  1^^6303). 

/.x=S 
or  -  1  +  CO  ^(11-1  +  -jV  V6303) 

+  o,^-^(lli-^iV6303), 
in  which 

0)  =  1 

or  -|(1±V3). 

4.    Find  the  cube  roots  of 

~-10  +  9i^3. 

Assume     i  (y  +  a  V^)  =  (-  10  +  9  i  V^)* 

=  a,^(-10  +  9V3),  (1) 

in  which 

and         (0  =  1  or-i(l+zV3)  or-|-(l-zV3), 
and  therefore 

(0^  -  1  or  - 1(1  -  ^V3)  or  - 1(1  +  i^S). 

.•.X(y_aV^)--o)^^(-10-9iV3).  (2) 


302  CUBIC   AND   QUAETIC    EQUATIONS. 

a)  +  (2),  y  =  a.^(-10+9/V3)+<o^^(-10-9;v3).  (3) 
(3)',  3/'  =  -20  +  21y. 

.•.(y-l)(2/-4)(y  +  5)  =  0. 

.••yi=l,    y2  =  4,    y3  =  -5.  (4) 

(1)X(2),   i(y'-2)  =  7. 

.•.z-2/^-28.  (5) 

.-.2,  =  -27,    Z2  =  -12,    Z3  =  -5.  (6) 

(1)~(2),   aVz  =  «)^(-10+9iV3)-«'^(-10-9iV3),  (7) 

(7)',  azy/z  =  18  J  V3  -  21  a  V2. 

,      18iV3 
••"^^=2lt^- 

Substitute  for  z  its  values  given  in  (6). 

.-.  av'2i  =  — 3iV3,  aVz2  =  2iV3,  aVz3  =  «V3-  (8) 

From  (4)  and  (8), 

(-  10  +  9iV3)J  =  i(l  -  3iV3), 

or  2  +  i-y/3, 

or  |(-5  +  iV3). 

5.    Find  the  cube  roots  of  4  +  iBi^5. 

Assume     ■^(y+aV2)  =  o)^(4+43iV5).  (1) 

■•.i(y-aV2)  =  <o'^(4-43iV5).  (2) 

(l)  +  (2),  2/  =  a,</(4  +  43^V5)  +  <«■^^/(4-48^•V5).  (3) 
(3)»,  2^  =  8+632/. 

•••(y-8)(/  +  8y+l)  =  0.  (4) 

.-.  Vi  =  8, 
y.  =  -4  +  Vl5.  (5) 

y3  =  -4-V15- 
(1)X(2),   i(y^--2)  =  21. 

.•.2=y^-84.  (6) 


CUBIC    AND    QUARTIC    EQUATIONS.  303 

Z,  =  -b3-8y/15,  (7) 

03  -  -  53  +  8  V15. 
(l)-(2),   aV2  -■=o>^(4  +  43iV5)-coV(4-432V5).   (8) 
(Sy,  az^z^  86zV5  -  63aV2. 

63  +  2; 
.\a^Zi=2i^5, 

^     86iV5     ^43^V5(5  +  4V15) 
"V^2     10-8V15  25-240 

--^(VS  +  4V3), 

Substituting  in  (1)  these  values  of  y  and  a-y/z, 
i(3/i  +  «V^i)  =  4  +  iV^, 
i(3/.  +  aV^2)-i(-4+V15-V^-4V3) 

i(y3  +  aV^3)  =  -i(l-^V3)(4  +  ^VS). 

Ex.  68. 

Solve  tlie  following  equations : 

1.  :r^  — 3:i;'  +  9:r-5--0.  7.  x^  +  3x^+x+l  =  0. 

2.  x^-3x^  +  9x-9  =  0.  8.   3a^^  +  27:r'^-9:i'+41  =  0. 

3.  2r'^~6a;2+18.r+17  =  0.      9.  Sx'  +  27x''-9x+4:=0. 

4.  :i^^-3^'-15a;-13  =  0.      10.  ;r^-18:^-33- 0. 

5.  x'~3x'-l5x~25=-0.      11.  n-^-9a;-12  =  0. 

6.  :^'  +  9:r'+9:r+15--0.         12.  x^~6x''  +  l0x-l=-0. 

13.  x'  +  y^  =  6,  and  :i;'  +  y'  =  l. 

14.  ^'^  +  y^  =  6  rn/n,  and  :r^  +  3/^  =  ?nn  (27?i  —  n). 


304  CUBIC    AND    QUARTIC    EQUATIONS. 

15.  x'-Sx'  +  9x  +  (k-8){l  +  Jc-')  =  0, 

16.  x'-Sx'--15x  +  {h  +  216)(l  +  Jc-')-200=0. 

17.  x'  +  9x'  +  9x-S(k+  24)  (1  +  k-')  +  16  =  0. 

18.  x^-2x-5  =  0. 

Find  the  cube  roots  of: 

19.  -8  +  6V3.  20.    -55-126iV3. 

21.    5  — 7iVS. 
Solve  ax^ -{-Shx'^  -{-  Sex  -{-  d=0,  given : 
22.    hd=c\        23.    2a'hd=  a'c^  +  h\ 

24.  a'bcd  =  a'c'  (b'  +  ac)  -  b\ 

25.  JcG  +  H^  =  ¥,  h  being  arbitrary. 

26.  Show  that  H=0  is  the  condition  that  ax'^  +  2bx-\-c 

shall  have  a  square  factor ;  and  that  A  =  0  is  the 
condition  that  ax^  -\-?>bx'^  +  Scx  -\-  d  shall  have  a 
square  factor. 

27.  Show  how  to  solve  the  cubic  by  assuming 

ax^  +  Sbx^  +  Scx+d 

=-m{ax  +  b  +  t,y  +  (l  —  m){ax  +  b  +  t,)\ 

and  determining  m,  ^i,  and  ^2- 

28.  li  ti=  (xi-\-  (00:2  +  oi^x^y  and  4  '=  (^i  +  w^^2  +  wr^s)^, 

where  w^  +  w  + 1  =  0,  find  z^i  +  4  and  z^i  2^2,  and  apply 
the  result  to  solve  the  cubic. 

29.  If  Xi,  X2,  and  x^  be  the  roots  of  a  cubic,  express 

(xi  —  x.2y(x2  —  x^y(x3--Xiy  m  terms  of  the  coeffi- 
cients. 

30.  Prove  that  if  all  three  roots  of  a  cubic  are  real  and 

unequal,  A  will  be  negative  ;  but,  that  if  two  of  the 
roots  are  complex,  A  will  be  positive. 


cubic  and  quartic  equations.  305 

§  53.   The  Quartic. 

Let  the  general  quartic  equation  be 

ax'  +  Ux^  +  ^cx'  +  ^dx  +  e  =  Q.  (21) 

Assume 

a{ax'  +  ^hx^  +  Gar'  +  4,dx  +  e) 

=  {ax''  +  2hx  +  c  +  2ty-  {2^rx  +  s)\  (22) 

Expand  and  equate  coefficients  of  like  powers  of  x. 

:.r  =  at~(ac~¥),  (23) 

s-yjr  =--2ht-{ad-  he),  (24) 

s''  =  ^f-\-4:ct-{ae-  c^).  (25) 

:.[at-  {ac  -  If)]  [4:f  +  4:ct-(ae-  c')] 

=  [2bt-(ad-bc)]\ 
.\4:f-~(ae-Ud+3c')t 
+  (ace  +  2bed-  ad'  -  eb'  -  c')  =  0,  (26) 

which  may  be  written 

4:t'-It  +  J=0,  (27) 

in  which  I=ae-4:bd+3 e\  (28) 

J=  aee  +2bed-  ad''  -  eb""  -  c\  (29) 

Selecting  any  one  of  the  three  values  of  t  determined  by 
the  cubic  (27),  the  corresponding  value  of  r  may  be  found 
by  substitution  for  t  in  (23),  and  then  that  of  s  by  substitu- 
tion in  (24),  or  if  r  —  0,  in  (25)  ;  and  the  quartic  in  (22) 
may  then  be  resolved  into  the  quadratic  factors,  « 

ax'  +  2{b-  -^r)x  +  e  +  2t-s, 
and  ax'  +  2{h  +  'y/r)x  +  e  +  2t-\-s.  (30) 

Each  of  these  factors  equated  to  zero  will  give  a  pair  of 
the  roots  of  the  quartic  equation  (21),  which  will  thus  be 
completely  resolved.  (31) 


306  CUBIC    AND    QUARTIC    EQUATIONS. 

The  equation  (27)  is  called  the  Reducing  Cubic  of  the 
Quartic  (21). 

Examples. 

1.    Solve  x'-Ux'  +  Alx'-QQx +  27  =  0.     (See  Ex.  41.) 
Let  x'  -  12^;^  +  ^7 x'  -  66x  +  27 

=  fx'^~6x+^  +  2 1\~  (2^rx  +  sf 

=  x'-  12a;^  +  /'se  +  ^+  4  ^5  -  iAx' 

-  (94.  +  2it  +  As^7^)  x+^  +  -^^t  +  4:e  -  s\ 
Equating  coefficients  of  like  powers  of  x, 
47  =  36  +  — +  4^-4r, 

66  =  94 +  24^5 +.4  5  V^, 
27  =  ^  +  -%U  +  Ae-s\ 

OD 


or 


and 


.■.6r  =  6t+7, 

s^r  =  -(6f  +  7), 

36  s'  =  144  i?  +  1128  i!  +  1237. 

.-.  (6^+7)  (144 <^+ 1128);+  1237)  = 

=  216(6i!+7)l 

.-.  6<  +  7=0, 

144  («+ 1128;!+ 1237  =  216(6  i!  +  7). 

.-.144;;'- 168^-275  =  0. 

.-.  (12i-25)(12<+ll)  =  0. 

.  ,  _  7    ,25       .  . 

11 
=  """12' 

r>         13 
.-.r,  =0,     '"2  =  -4'       »"3" 

1 
=  4' 

si'  =  i^.    s.,  =-3V13,  S3-- 

=  -3. 

CUBIC  a:nd  quaktic  equations.  307 

n  ,      IIV         13 

^  2j       4 
=  (x'~6x+  I2f  -  13  (^  -  3)^ 

=  (x'-6x+6y-(x-  sy  =  0. 

The  last  gives 

:?;'  — 7a;  +  9=-0    or  :r' -  5:r  +  3  =  0. 
.\x^-^(1  ±:-y/lS)  or  ^  =  l(5=fc  VIS)- 

2.    Solve9.^*-54:i;H60^'-72:z;  +  16  =  0. 

Here        a -9,  b  =  -?^,  c=10,  d=~lS,  e  =  16. 

.'.  r-=9t-  (90  -  182i)  =  2i(4:t  +  41), 
5  V^  -  -  27  ^- (- 162  + 135)  -  -  27  (^^  -  1), 

=  4:(f  +  10t-  U)  =  4:(t  +  ll)(t-l). 

.'.9(4:t  +  41)(t  +  ll)(t~l)=.129(t-l). 

or  (4^  +  41)(^+ll)-81(^-l). 

If  ^-1,  .-.r-lOli  and  5  =  0. 

.-.  (9x''  -  21 X  +  12)^  -  405a;^  =  0. 

.-.  3a;^-(9  +  3V5)^-  +  4  =  0, 
or  ^x^  -  (9  -  3 V5)^  +  4  =  0. 

.-.  x  =  l[9  +  3 V5  ±  V(78  +  54 V5)], 
or  x=^i[9-  3 V5 ±  VC^S  -  54 V5)]. 

Ex.  69. 

Solve  tlie  following  equations  : 

1.  x'-Qx^~2x''  +  ?>Qx-2^  =  Q. 

2.  :^*-2a;^-25^'  +  18:x;  +  24  =  0. 


308  CUBIC    AND    QUARTIC    EQUATIONS. 

3.  2:r*-5:r'-17:^'  +  53:r-28-30.     5.  :r*- 12a;- 5  =  0. 

4.  x'+14:x'  +  4:8x  + 4:9  =  0.  6.  a;*-12:r-17-:0. 

7.  x'-8x'-12x'  +  S4:x-6^  =  0. 

8.  x'  +  2x'-S7x'~S8x+l  =  0. 

9.  121r^*  +  198a;^-100^^-36a;  +  4-=0. 
10.  x'+y^li,  x  +  y''  =  2i. 

§  54.  The  cubic  in  (27)  will,  in  general,  give  three  values 
of  t.  Let  them  be  denoted  by  ti,  t^,  4-  Also  let  the  corre- 
sponding values  of  r  and  of  s  be  denoted  by  ri,  r^,  r^,  and 
Si,  52,  53,  respectively.  Let  Xi,  x^,  x^,  and  x^  denote  the  roots 
of  the  quartic.     Then,  by  (31), 

ax''  +  2(b-  ■^n)x  +  c  +  2t,-s,  =  0  (32) 

will  furnish  a  pair  of  the  roots  of  the  quartic  (21),  say  a;i,a;2, 

^^^  ax'  +  2(b  +  ■^r,)x  +  c  +  2t,  +  s,  =  0 

will  furnish  the  complementary  pair,  x^,  x^. 

So  also  ax''+2(h  —  ^r,)x  +  e  +  2^  -  ^2  =  0  (33) 

will  furnish  a  pair  of  roots  different  from  either  of  the  above 
pairs,  say  Xi,  x^,  and 

ax'  +  2{b+  ^r,)x+c  +  2t,  +  s,  =  0 

will  furnish  the  complementary  pair  X2,  x^. 

Finally, 

ax'  +  2(b  -  ^r,)x  +  c+2t,  -s,  =  0 

will  furnish  a  pair  of  roots  different  from  any  of  the  pre- 
ceding pairs.  These  must  therefore  be  either  Xi^  x^,  or  else 
X2,  x^.     Then 

ax''  +  2(b  +  ^r,)x+c  +  2t,  +  s,  =  0 

will  furnish  the  complementary  pair ;  that  is,  either  x^,  x-s, 
or  Xij  x^,  as  the  case  may  be. 


CUBIC   AND    QUARTIC    EQUATIONS.  309 

Let  y^  =  1,  then  y  may  be  so  determined  that 

ax'  +  2{h-y^o^,)x  +  c  +  2U-ys,^0       (34) 

will  furnish  the  pair  of  roots  Xi,  x^,  and  then 

ax'  +  2(b  +  y-yyr,)x  +  c  +  2^  +  753  =  0 

will  furnish  the  complementary  pair  X2,  x^. 

■Bj(32),x,  +  x,  =  ~^(b-^n). 
By  (33),  x,  +  x,  =  ~-(b-^r,), 

CL 

By  (34),  x,  +  x,  =  -^(b  ~y  Vra).  (35) 

45  t 

By  (21),  x^  +  x^  +  x^  +  x^  = 

a 

.'.  axi  +  b  =  v^i  +  V^'2  +  r  V^3 ; 

ax2  +  b  =  Vri  —  Vra  —  y  -y/r^ ; 

arr3  +  Z>  =  —  Vn  +  V^2  -  r  V^3 ;  (36) 

ax^  +  b  =  ~-  Vn  -  ^n  +  r  V^3. 

Also,  from  the  first  three  equations  of  (35), 
7  V^^i  V^2  V^'3  =  [^+2  a(a;i+:r,)][5+ Ja(^i+^3)][5+^  a(:?;i+:r4)] 
=  P+iab'(Sx,+x,  +  x,  +  x,)  +  ia'b[(x,+x,)(x,  +  x,) 
+  (^1+^3)  (^i+^4)  +  (^i  +  ^4)  (^1+^2)] 
+  i  a^  (xi + 3:2)  (:?;i + x^)  (xi  +  rr^) 
==:  F  +  iab\Sxi  +  x,^  +  X3  +  x^) 

+  ia'5  [3^i'  +  2^1  (^^2  +  ^-3+^^4)  +^2^3  +  ^^2^4+^3^'4] 

+  I  a^[xi^  +  x;'  (x^2+^3+Xi)+Xi(x2X^+x\Xi+x^x,)+x.,x,x,] 
^53+ia5^(2^,-l^)  +  ia^5(2.,^-^^,+^^) 

+  ia'(-  —  x,'-~)==~b'+^abc-ia'd 
\      a  a  J  ^ 

=  -i(a''d-Sabc  +  2b'), 


310  CUBIC    AND    QUARTIC    EQUATIONS. 

That  is,    y-yJn-yJi\^n  =  -iG.  (37) 

[See  (5),  page  297]. 

Therefore,  y  =  +  1  or  -  1  (38) 

according  as  G  is  negative  or  positive. 

Hence,  by  (4),  (5),  (21),  (23),  (27),  (28),  (29),  (36),  and 
(38),  if        ax'  +  4  bx'  +  6  ex'  +  Adx  +  e^O, 
then  shall 

Xi=-(-b  +  V^"i  +  V^^2  +  r  V^'3), 

x^=-(~b  +  v^'i  —  V^'2  —  y  V^s), 

Xs=-(-~b~  y/r,  +  V^2  -  y^/r^),  (39) 

x^  =  -(~b  -  Vn  —  V^'2  +  y  V^'3), 

in  which      ri  =  ati  —  H, 

T^^at,-R,  (40) 

r^  =  at^  —  H, 
ti,  4,  4  being  the  roots  of  the  equation, 

4:f-It+J=0,  (41) 

JH:=^  ac  -  b\ 
I  ^ae-Ud+?>c\ 
J  =  ace  +  2  bed  —  a(^^  —  ^^^  —  c^, 
Q=a^d-Zabo-\-'lb\ 
and  y  =  +  1  or  —  1  according  as  G  is  negative  or  positive. 

§  55.    The  roots  given  in  (39)  may  also  be  expressed  in 
terms  of  any  one  of  the  three  values  of  t,  as  follows : 

By  (40),  ri  +  r,  +  r3-a(?!i  +  «52  +  4)-3^ 


CUBIC    AND    QUARTIC    EQUATIONS.  311 

But  (41),  t^  +  t,  +  t^  =  0. 

.•.n  +  r2  +  r3--3^. 
Now,       v^2  +  r  V^3  =  V(^'2  +  ^3  +  2y  V^2  V^3) 

=  V(-  n  -  3  ^ ^\  See  (37). 

In  like  manner  it  may  be  sliown  that 

Eeplacing  r^  by  ati  —  ^,  see  (40),  and  solving  (41),  the 
result  becomes 

If  ax^  +  ^hx^  +  ^cx"  +  ^dx  +  e  =  0, 

then 

..  =_1 1  -._V(«^-^)  f  ^[-«^-2^+;;^J  }  .(42) 


in  whicli      t  =  -  i^'d  ^+  ^,^3  V(27/^- P)] 


SML"    '  3V3 


■i\l[^-3^v(^^'^-^>]      (^^) 


ZTr::.  ac  -  h\ 

I  =^ae-Ud+Zc\ 

J  —  ace  +  2  hcd  —  ac?^  —  e^^  —  c^, 

G=a^d~?>ahc+2h\ 


312  CUBIC    AND    QUARTIC    EQUATIONS. 

Ex.  70. 

1 .  Keduce  the  quartic  ax^  -{-  4:bx^  -{-  6  cx"^  ~\-  4:  dx  -\-  e  =  0 

to  the  form  y'  +  6irf  +  4:G7/+  a' I-  ^II'  =  0. 

2 .  Show  that  the  two  quartics  x^  +  6  Hx^  =b  .4  (?^  +  K^  0 

have  the  same  reducing  cubic. 

Solve  the  quartics  : 

3.  r^^  -  24a;' ±32:r- 132-0. 

4.  a;^-6:^'±208a;-321-=0. 

5.  rr^-e:^;' =4=  16:^  —  33=^0. 

6.  x''-^x^±:\^x-\-m  =  ^. 

7.  :r*-6:r'±48:?;— 117==0. 

8.  :t^*  + 6^'  — 60^  +  36  =-0. 

Show  that,  if  x^,  x^,  ^3,  x^  be  the  roots  of  a  quartic, 

9.  A.^H=--%{x^-xi)\ 

10.    '2.^aU^%{x,-x^\x^-x:)\ 

11.       oA  (jT  ^^^^  ztz  {^Xi~\-X2       X^     ~^4)\^1\'^3      ^i      ^2yV'^l~l  *^4      •^2       "^S/- 

12.  432  a^J^  [(xi  —  x^)  (x^  —  x^)  —  (x^  —  x^)  (x^  —  ^^2)] 

X  [(xi  —  Xs)  (^4  —  ^2)  —  (^1  -  ^4)  (^^i  -  ^^3)] 
X  [(xi  —  x^)  {x^  —  Xs)  —  (xi  —  x^)  {x^  —  X,)]. 

13.  4.ir'-a'Iir+a'J+G'  =  0. 

14.  Prove  that,  if  27J'^  =  /^  the  quartic  has  a  pair  of 

equal  roots. 

15.  Prove  that,  if  /=  J=0,  the  quartic  has  three  equal 

roots. 

16.  Prove  that,  if  a^I=12JI^,  and  a^J=8IP,  the  quartic 

has  two  distinct  pairs  of  equal  roots. 


CUBIC    AND    QUARTIC    EQUATIONS.  313 

Solve  the  quartic  x'  +  G  Ha:'  +  4  Gx  +  ci'I-  SH'^O: 

17.  By  reducing  it  to  the  form 

(x'  +  2^7/x  +  ^0  (x'  -  2^yx  +  z,)  =  0. 

18.  By  reducing  it  to  the  form 

y2  (^'  +  yix  +  zf  -  yi  {x'  +  y.,xy  =  0. 

19.  By  reducing  it  to  the  form 

{x'J^yy~{z,x-\-z,J  =  ^, 

20.  By  assuming  the  roots  to  he  of  the  form 

in  which  a^  =  1,^^  =  1,  y^-=l. 

21.  By  assuming  the  roots  to  be  of  the  form 

in  which  z"-^  +  1  =  0,  and  n  is  integral. 

22.  Apply  the  method  of  Exam.  20  (Euler's  Method)  to 

solve  the  quartics 

x'-  6  Ix^  ±  8  V(^+  ^?^'+  ^'-3  Imn)  :r-3(4  mn  -I')  =  0. 

23.  Reduce  the  quartic  to  the  form  y*+  6  Cy^  +  ^  —  0,  by 

assuming  x=  ^        ''^^,  and  suitably  determining  z^ 
and  Z2. 

24.  Reduce  the  quartic  to  the  form 

y*  +  4  %'  +  6  (7/  +  4  %  + 1  =  0 , 

by  assuming  ^  — 2:1  +  ^2^,  and  suitably  determining 

Zi  and  Z2. 

25.  Make  the  same  reduction  as  in  the  last  question,  by 

assuming  x  =  Zi-{~ z^y'^,  and    suitably    determining 
Zi  and  z-2. 

26.  Eliminate  x  between  x'+6irx''  +  AGx+a'I-Sir''  =  0 

smd  x'^-{-2yx-{-z^=0,  and  so  determine  y  that  the 
resulting  equation  may  reduce  to  the  form 
z'  +  e>Cz''  + U==0. 


314  CUBIC   AND    QUARTIC    EQUATIONS. 

If  Xi,  X2,  ^3,  ^4  denote  the  roots  of  the  quartic 
x'  +  6JIx'  +  4.Gx  +  a'I-3IP  =  0, 
form  the  cubic  whose  roots  are  : 

iOO.      XyX^  "I"  *^3'^4)    ^1^3     1"  X^X^^    X^X^  ~\~  x^x^.  ^ 

00  ^1^2  ^3^4  ^1*^3  X4a;2  XiX^  ^2**^3 

Xi  ~Y'  X2        X^      ■  X^       Xi  ~p  X^         X^      ■  X2       X\  ~Y'  X^        X2        X^ 

30.    (X1X2  —  ^3^4)  (^1  +  ^2  —  ^3  —  ^4),  etc. 

31.  (x^-x^y^x^—x^y,  (xi—x^y^x^—x^y,  (xi-x^y^x^-x^y, 

32.  (xi  —  X2)  (X2  —  Xs)  (x^i  —  Xi)  (Xi  —  ^1), 

(^1  ^3)  V^3  ^i)  (.^4  ""  ^2J  V'^2  ^1/j 

(:ri  —  :r4)  (0:4  —  X2)  (X2  —  x^)  (x^  —  ^1). 

33.  Show  how  to  solve  the  quartic,  knowing  the  roots  of 

any  of  the  above  cubics. 

34.  Reduce  each  of  the  cubics  in  Exams.  27  to  32  to  the 

standard  form  Ay^  +  Cy  +  B^O. 

Form  the  equation  whose  roots  are  : 

35.  The  squares,  36.    The  cubes, 

of  the  roots  of  ax^  +  3  bx'^  +  3cx  -{-  d=0. 

Form  the  equation  whose  roots  are  : 
37.    The  squares,  38.    The  cubes, 

of  the  roots  of  ax''  +  Ux^ +  6cx^ +  4:dx  + e  =  0. 

39.  Form  the  equation  whose  roots  are  the  squares  of  the 

differences  of  the  roots  of  a  cubic. 

40.  Form  the  equation  whose  roots  are  the  squares  of  the 

differences  of  the  roots  of  a  quartic. 


CHAPTER    X. 

Determinants. 


I.   Definitions  and  Notation. 
§  56.    The  symbol  ai  h^ 

denotes  the  expression     aiZ>2  —  a^x, 
which  is  called  a  Determinant  of  the  Second  Order. 
The  symbol 


denotes  the  expression 

a,    ^2  (?2    —a-i 


«1 

h 

Ci 

^2 

b. 

C2 

^3 

k 

C:i 

bi  Ci 

^3     ^3 


+  «3  I   ^1     Ci 

\  h  c. 


which  is  called  a  Determinant  of  the  Third  Order. 

The  symbol 


a. 

h, 

Cl 

h 

(22 

K 

C2 

^3 

h. 

Cz 

h 

denotes  the  expression 


«! 


^2     C, 

h 

-^2 

h     ^1 

h 

bs  Cs 

^3     ^3 

h 

K    Cn 

K 

hn    Cn 

"k 

+  ---+(-l)"^'a„ 


h 


-1  f^n- 


which  is  called  a  Determinant  of  the  nth  Order. 


316 


DETERMINANTS. 


3. 


a~\-h     a  —  h 
a  —  h     a-{-h 


Examples. 
=  {a  +  hy  ~  {a~hy  =  4:ah, 


\  X     y 
1  x^    y' 

1  a;"  y' 


T    y' 
x^'  3/" 


X    y 
U"  y" 


+ 


X    y 
x^  y^ 


-  x^y^^  —  x^^y^  —  xy^^ + x^^y  +  xy^ — x^y. 


12  3 

=  13  4 

-2 

2  3 

+  3 

2  3 

2  3  4 

4  5 

4  5 

3  4 

8  4  5 

- 1(15-16)-2(10-12)  +  3(8-9) 
-l+4-3-=0. 


^  a  h  c 

a  0  z  y 

b  z  0  X 

c  y  X  0 


=  —  a 

a  h  c 

+  h 

a  h   c    —c 

a  h  c 

z   0  X 

0  z   y 

0  z  y 

y  X  0 

y  X  0 

z   0  ri 

=  +  a  V   —  abxy  —  cazx 

4"  ^y   —abxy   —hcyz 

-hcyz    — cazx 


•  gy 


—  a  V + Z^y  +  c^2;^  —  2  abxy — 2  ^(^y^;  —  2  caz^r 


Ex.  71. 

Expand  the  following,  i.e.,  write  them  in  ordinary  alge- 
braic notation : 


1. 


a  y 

2. 

a  b 

^,      y  a 

4. 

b   a 

b    X 

y  ^ 

X    b 

X  y 

ma  y 
mh  X 

6.      ma  my 

b      X 

7.      a  m.y 

b    7)IX 

8. 

a      y 
mb  mx 

a  +  ' 
h  +  ' 

/  y 

V    X 

] 

10. 

a  + 
I 

mh  y  +  m,x 

)                   X 

11. 

a,  - 
a\- 

-b 
~b' 

DETERMINANTS. 


31^; 


12. 


x"^  +  a^       ah 
ah      x"^  +  I 


13. 


ab 
-a' 


-be 


14. 


a  —  b,  —  2a  I 

~2b,b~a\ 


15. 


C(: 

a^ 

«3 

b. 

^. 

^3 

Ci 

Ci 

^3 

16. 


Ch 

h 

Cs 

a^ 

^2 

Ct 

a, 

b. 

Ci 

17. 


X  y  z 
z  X  y 
y  z   X 


18. 


1 

1 

1 

X 

y 

2 

a 

b 

e 

19. 


1  1 

1 

a    h 

e 

\a^  b' 

c' 

20. 


a 

X 

y 

:?; 

a 

z 

3/ 

z 

a 

21. 


a      b  -^c      a 
b  b      c-\-a 


22. 


1  1 

Y+x      1 

1       1+2/ 


23. 


mai  m^i  m(?i  I 
a,       ^2       '^•i 

as         ^^3         ^3 


24. 


ai  +  7/?.a3  b^  +  mb^  e^  +  mc^ 
a.i  h,  ^2 


r/3 


^3 


25. 


«1  ^1 

e^  d^ 

^2     ^2 

<?2     C?2 

«3     ^3 

(?3     0?3 

a,  b^ 

^4     6?4 

26.       «!  a2  '^3  ^4 

I  b^  h,  b.,  b^ 

C\  ^2  <^3  ^4 

I  di  d^  d-i  di 


29. 


27. 

1111 

abed 
a?  P  c'  d' 
a'  P  c'  d' 

28. 

X       0 

0     -y 
0       0 

0     a, 
0     a, 

X        «! 

-y  ao 

1       1 

1   1  + 
1       1 
1       1 

1          1 

X       1            1 

1      14 

-  2 

30. 

1-J 

1 
1 

-a      1 

1  +  ^ 
1 
1 

1 
1 

1  +  ^ 
1      1 

1 
1 
1 

+  d 

§  57.  The  quantities  whicli  in  the  determinant  notation 
stand  unconnected,  and  which  are  taken  as  factors  to  form 
the  terms  of  the   expanded    determinant,   are    called  the 


318  DETEEMINANTS. 


Elevients  of   the    determinant ;    e.g.,   the    elements  of  the 
determinant 

a'\-h      a  b 

c      c  +  d      d 
e         f      e+f 

are  the  nine  quantities  a-{-h,  a,  h,  c,  c~{-  J,  d,  e,f,  e  +/. 

The  elements  standing  in  any  horizontal  line  constitute 
a  Row,  and  those  standing  in  any  vertical  line  constitute  a 
Column.  The  rows  are  numbered  first,  second,  third,  etc., 
beginning  at  the  top,  and  the  columns  are  similarly  num- 
bered, beginning  on  the  left.  The  elements  of  a  row  are 
numbered  first,  second,  third,  etc.,  beginning  on  the  left, 
and  those  of  a  column  are  similarly  numbered,  beginning 
at  the  top.  Hence  the  Tuih.  element  of  the  nth.  column, 
called  the  (m,  n)th  element,  is  the  nth.  element  of  the  mth 
row;  it  is  often  denoted  by  a^^„,  the  first  suffix  denoting 
the  row,  and  the  second  the  column  to  which  the  element 
belongs.     Thus,  in  the  above  determinant, 

«2, 3  "=  d,    a^^ 2  =fi    a2,2  =  c  +  d. 

A  determinant  is  said  to  have  two  diagonals,  called,  re- 
spectively, principal  and  secondary.  The  elements  stand- 
ing in  a  line  from  the  upper  left-hand  corner  to  the  lower 
right-hand  corner  constitute  the  principal  diagonal ;  those 
standing  in  a  line  from  the  upper  right-hand  corner  to  the 
lower  left-hand  corner  constitute  the  secondary  diagonal. 
Thus,  in  the  determinant  given  above,  the  principal  diag- 
onal elements  are  a-\-h,  c-\-  d,  e  -\-f,  and  those  of  the 
secondary  diagonal  are  h,  <?.+  d,  e. 

The  product  of  the  elements  standing  in  the  principal 
diagonal  is  called  the  principal  or  leading  term  of  the 
determinant. 

Where  there  is  no  danger  of  ambiguity,  a  determinant  is 


DETERMINANTS. 


319 


often  denoted  by  writing  only  its  principal  diagonal  ele- 
ments between  vertical  bars  or  within  parentheses.     Thus, 


ai  bi  Ci 

(2.2     ^2     <^2 

as    h    <?3 

a     X    y 

a'  x^  y 

a"  x^^  y" 

^5    ^5    h 

c^  92  h 

Co  go  h 

I  ci\  ^2  <?3 1  denotes 


I  a  x'  y'^  I  denotes 


and  I  ^5  ^2  h  I  denotes 


If,  in  any  determinant  of  order  n,  the^th  row  and  the 
qth  column  be  erased,  and  all  the  rows  above  the  ^th  and 
all  the  columns  to  the  left  of  the  ^'th  be  transferred  in  order 
over  the  others,  the  resulting  determinant  multiplied  by 
(—  l)("-i)(^+^)  is  called  the  complement  of  the  (p,  q)th  ele- 
ment. 

If  n  be  odd,  (-  l)(-i)(^+^)  =  1  for  all  values  of  ^  +  ^. 

If  the  elements  of  any  determinant  are  each  denoted  by 
a  small  letter,  and  are  all  different,  the  complement  of  any 
element  may  be  denoted  by  the  corresponding  capital  letter 
affected  with  the  suffix  or  suffixes  of  the  element. 

Thus,  in 


ai 

h 

Ci 

^2 

b. 

Ci 

as 

b. 

^3 

A^  denotes  the  complement  of  a^,  which  is    b^  c^ 

bz  c-i 

A2  denotes  the  complement  of  ^2,  which  is  I  b^  c^ 


B^  denotes  the  complement  of  ^3,  which  is 
In  these  n  —  1  -  2,  and  .-.  (-  l)(-i)(^+^)  =  1. 


Ci  ai 
C2  a^ 


320 


DETERMINANTS. 


In 


A,  =  ~ 


and  (7q 


h 

C:, 

d. 

b. 

Ci 

d, 

bi 

Cx 

d. 

d. 

a. 

h 

d. 

«! 

b, 

d. 

a^ 

b. 

<2i  bi  Ci  di 

a^  ^2  ^2  <^2 

a^  hs  c,  ds 

a^  h^  c^  d^ 


Here  92  —  1  =  3,  p  =  1,    ^  =  2, 

and  .-.  (-l)(-i)(i'+.)  =  _i. 

Here  n  —  1  =  3,  J9  -^  3,    ^  =  3, 

and  .-.  (-  l)(-i)0'+^)  ==  1. 


Written   in    the    diagonal    notation,    the    last    example 
would  be  : 


In  I  ai  Z>2  <?3  <^4 1     ^2 


i  ^3  <^4  <^i  I    and   C^  =  \d4,  a^  h^ 


To  find  the  complement  of  a  product  of  two  or  more 
elements,  find  first  the  complement  of  one  of  the  elements ; 
then  in  this  complement  find  the  complement  of  a  second 
element  of  the  product ;  next,  in  this  second  complement 
find  the  complement  of  a  third  element  of  the  product ;  and 
proceed  thus  through  the  whole  product.  The  final  com- 
plement will  be  the  one  required. 

For  example,  to  find  the  complement  of  aj^s,  first  find 
A2 ;  then  in  A^  find  the  complement  of  h^.  Similarly,  to 
find  the  complement  of  aj)ids,  first  find  ^4 ;  then  in  A^  find 
the  complement  of  hi ;  then  in  this  complement  find  the 
complement  of  d^.  The  order  in  which  the  partial  comple- 
ments are  found  will  affect  the  form,  but  not  the  value  of 
the  result.  Thus  the  expansion  of  the  complement  of  aj)ids 
will  be  the  same  whether  found  in  the  order  a^,  hi,  d^,  or  in 
the  order  a^,  d^,  hi,  or  again,  in  the  order  d^,  hi,  a^. 

The  TTith  element  of  the  nth  column  is  called  the  con- 
jugate of  the  nth  element  of  the  mth  column,  and  vice  versct ; 
^•^•,  <^m,n  and  an,m  ^re  each  conjugate  with  respect  to  the 


DETERMINANTS.  321 


other,  or  are  a  pair  of  conjugates.  Each  element  of  the 
principal  diagonal  is  its  own  conjugate  or  is  self-conjugate. 

A  symmetrical  determinant  is  one  in  which  each  element 
is  equal  to  its  conjugate. 

A  skew  determinant  is  one  in  which  the  sum  of  each  j9aiV 
of  conjugates  is  zero. 

A  skew  symmetrical  determinant  is  a  skew  determinant 
,     whose  principal  diagonal  elements  are  all  zeros. 

Ex.  72. 

Write  in  the  ''  square  "  notation  : 
1.    I  ai  ^3  ^5 1  2.    I  x^  yi  23 1  3.    I  a^  c^  e^  \ 

4.    \wQX^y^z^\     5.    I  aj,  hi  c^  d^\     6.    |  a^^  j  a.,,  2  cLz,z  a^,i  | 

In  I  ai  ^2  c-i  c?4  65 1  find  the  complements  : 
7.    A,.       8.  B,.      9.   B,.       10.   A.       11.    Q.       12.   U,, 

In  the  same  determinant  find  the  complements  of : 
13.  a^b^.    14.  ajC^.    15.  a^Ci.    16.  aih^Ci.    17.  b^d^ei.    18.  a^c^d^. 

Prove  that : 

19.  I  ai  ^2  <^3 1  =  <^i^i  +  ^1^1  +  C\C^  =  aiAi  +  aa^z  +  ^s^s- 

20.  a,A,  +  b,JB,  +  c,C,=^  a^A,  +  b,B,  +  c,C, 

=  aiJBi  +  (22^  +  ^-^s  ^  «iQ  +  CI2O2  +  (23(73  =  0. 

II.    Transformation. 

§  58.  Theorem  I.  The  value  of  a  determinant  luill  not  be 
altered  if  the  columns  be  written  in  order  as  rows^  and  vice 
versa. 

Hence,  in  any  theorem  in  which  the  word  ''row  "  occurs, 
the  word  "  column  "  may  be  substituted  therefor,  and  vice 


322 


DETERMINANTS. 


versa;  and,  in  any  theorem  in  which  both  ''row"  and 
"  column  "  occur,  these  words  may  be  interchanged  without 
aJffecting  the  truth  of  the  theorem. 

.  Theorem  II.  If  any  two  rows  {or  two  columns)  of  a 
determinant  he  interchanged^  the  resulting  determinant  will 
differ  only  in  signfro'tn  the  original  one. 

Cor.  1.  If  a  row  {or  a  colum^n)  he  transferred  over  n 
other  rows  {or  columris),  the  determinant  will  he  multiplied 

Cor.  2.  A  transfer  of  p  consecutive  rows  {or  columns) 
over  m  —  p  other  consecutive  rows  {or  columns)  multiplies 
the  determinant  hy  {  —  iy°»-i)p^ 

Cor.  3.  If  Ap,q  denote  the  complement  of  the  (p,  q)^A 
element  of  any  determinant  A  of  order  n,  and  \^^  denote 
the  determinant  forTYied  from  A  hy  strilcing  out  the  ]^th  row 
and  the  qth  column,  then  will  Ap^  q  —  (~  l)^^'^^Ap^  q. 


Examples. 


ai  hi  Ci 
a<i  h^  C2 
as  h,  c. 

+ 

ai  ^2  <^3 
hi  ^2   h 

Ci     C2      Cs 

h^  hi  ^3 
a^  ai  ag 

C2      Ci      <?3 

hi  ai  Ci    =  — 

^2     <^2     <^2 

hs  aa  c^ 

hi    ^2    ^3 
«!     ^2     <^3 
Ci     C2     Ci 

2.  Transform  [a-i  h^  c-s  \  so  that  h^c^a-i  shall  be  the  first  row. 

ai  hi  Ci    =   a^  hs  c-^    ==    h.^  c-^  a^ 

a2  ^2  ^2  ai  hi  Ci  hi  Ci  ai 

as  hs  Cs         a2  ^2  ^2  ^2  C2  ^2 

or     ]ai  h2  Cs\  =  \as  hi  C2\  =  \  h^  Ci  02  \ 

3.  Transform  |  aj  ^2  c^  \  so  that  J2<^3^i  shall  be  the  principal 

diagonal  elements  in  order. 

I  <^l     ^2     ^3  I  =  I  «2     ^3     ^1  [  =  —  I  ^2     «3     Ci  I 


DETERMINANTS. 


323 


4.    If  A  ==^  I  ai  ^2  ^3  <^4  e-^f(i  I,  find  Aa^g  and  ^3,5. 
Here      a^^^^  e.^\    -'-  ^z,^  =  \a^h.,  c^  d^f^\ 
and       ^3,5  =^  (-  lY-'^^'^'^  I/,  ^5  5e  c,  d,  I 


Ex.  73. 

Transform  a  h  c 

def 
g  h  k 

into  an  equivalent  determinant  having : 

1.  cf,  h,  k  2^^  its  first  row. 

2.  b,  e,  h  SiQ  its  first  row. 

3.  d,f,esiS  its  first  row. 

4.  /,  ^,  6?  as  its  first  row. 

5.  c,  e,  g  SiS  its  principal  diagonal. 

6.  g,f,hsiS  its  secondary  diagonal. 

Transform  abed 

e  f  9  ^ 
k  I  m  n 
p  q   r    s 

into  an  equivalent  determinant  having  : 

7.  a,  d,  c,  b  as  its  first  row,  and  a,  k,  e,  p  as  its  first 

column. 

8.  5,  n,  h,  d  as  its  third  row,  and  m,  I,  n,  k  as  its  second 

column. 

9.  e,  TYi^  q^  d  as  its  principal  diagonal. 
10.    5,  ??^,/,  a  as  its  secondary  diagonal. 

Prove  that : 

11.      a  b  0    c     -=   g  0  0  0 

I  111  J  I  k 

0  c  a  b 

f  0  d  e 


a  b  0    c 

r= 

d  e    f   0 

0  0  (J   0 

h  k  I   m 

324 


DETERMINANTS. 


12. 


13. 


0   a   b  0 

= 

c    d  e  f 

g   h   Ic  I 

0  7)1  n  0 

d 

c  j    e 

a 

0  0  h 

m 

0  0  n 

h 

9  ^   ^ 

d  a   c  h 

rzr 

10   0  0 

g^  f  e 

k  0  hO 

a 

0  0  0 

b 

e  0  0 

c 

/  A  0 

d 

9  fc  I 

a 

b 

c 

d 

b 

c 

d 

a 

c 

d 

a 

b 

d 

a 

b 

c 

Transform 


so  as  to  have  tlie  principal  diagonal  composed  of : 
14.    The  four  a's.  15.    The  four  5's. 

16.    The  four  c's.  17.    The  four  c^'s. 


Prove  that : 


18. 
19. 
20. 
21. 
22. 
23. 


a. 


h\^-\h. 


ai  ^2  ^3  ^4 1  —  I  di  C2  h 
ai  h^  C'i  d^  65  |="  I  ^1  ^2 


ai  h^  C3  d^  ^5  /e  I  =  —  |/i  ^2  4  c^  h  a^  \ 
If  two  determinants,  A  and  A',  of  the  Tith  degree  be 
such  that  the  first  row  of  the  one  is  the  same  as  the 
last  row  of  the  other,  the  second  row  of  the  one  the 
same  as  the  (?2  — l)th  row  of  the  other,  the  third 
row  of  the  one  the  same  as  the  (n  —  2)th  row  of  the 
other,  and  so  on,  then  will  A  =  (— l)i^("-'^A'. 

Transform,  by  cyclic  transposition  of  the  rows  and  col- 
umns, the  determinant  |  a^  h^  c^  di  e^]  into  an  equal  deter- 
minant having : 

24.    c^  in  the  first  row  and  first  column. 


DETERMINANTS. 


325 


25.  d^  in  the  first  row  and  first  column. 

26.  <?3  in  the  second  row  and  fifth  column. 

27.  e^  in  the  third  row  and  second  column. 


Given  A  — |ai  Z>2  c^  d^  e-^  f^  gi\,  determine: 


28.    A,,  6. 
32.    A. 


29.    A,,  5. 
33.    A,,,. 


30.    A, 


34.    A. 


31. 
35. 


2,6-  •^*^*      -^3,  5-  *'■*•      ^^3,7-  *^»^*      -^5,  2- 

36.  Prove  that  a  determinant  will  not  be  changed  in  value 
by  any  permutation  of  the  rows  and  the  columns 
which  merely  changes  the  order  of  the  elements 
of  either  diagonal,  without  changing  the  elements 
themselves. 


§  59.  Theorem  III.  If  A  denote  any  determinant  of  order 
n,  ^^^^  the  determinant  formed  from  A  hy  strilcing  out  the 
ipth  roiu  and  the  qth  column,  and  a^^  q  the  (p,  q)th  element 
of  A,  then  will 


:(-l)^-^K,Ai,,-a2,,A,, 


^  + +  (-1)^ 


a, 


Cor.  1.   A=:ai,^J.i,^  +  a2,^^2,3+«3,^^3,3+ 


'p,^^p,q\    J- 


-  a^^  \Ap^  1  -j-  ap^  2-^i>,  2  r  (^p>  3^p 


i  + +  ««,n^ 


Examples. 


1.    Let  A  ~  I  ai  h^  c-s  \  and  q  =  S. 


a,  h,  Ci 

Ci 

a,  h,  1 

^2     1>-1     C'l 

--^{-ly 

C2  a^  Z>2 

a-,  h  c. 

c-i  a^  b's 

{-~iy{' 

1       ^2     h       ~ 

-C2 

cti  h, 
a.,  b-s 

+  c. 

ai  bi 
^2    ^2 

=  (— l)XeiAi,3  -  C,A2,3  +  ^3^3,3). 


326 


DETERMINANTS. 


2.    Let  A  =  I  ai  Z)2  ^'3  (i^  e-^  |  and  ^  =  4. 
.-.  A==(— 1)'|  c?i  a2  ^3  c,  e-^\ 

"=■'■  (—  1)'  (<^l  I  ^2  ^3  6'4  ^5  i  —  C?2  I  «!  ^3  <:?4  ^5  I 
+  (^3  I  «!  Z)2  <:?4  ^5  I  -  ^4  I  Cbl  h  C3  <?5  | 
+  C?5  I  «!   ^2   C3  ^4  I  ) 


Cor.  2.     If  the  elements  of  the  ^th  row  all  vanish  except 
the  qth,  then  shall  A  ==  (  -  1)^+%,  ^\,^. 


Example. 


a 
d 
h 

h  0  c 

e  f  9 
h  0   I 

m 

n  0  p 

:(_1)>+^ 


/ 

d 

e 

9 

0 

a 

b 

c 

0 

h 

h 

I 

0 

m 

n 

P 

=(-iry 


a  b  c 
h  h  I 
7)1  n  p 


Cor.  3.  If  the  elements  on  one  side  of  the  principal  diag- 
onal of  a  determinant  be  all  zero,  the  determinant  will  be 
equal  to  the  product  of  the  diagonal  elements. 

If  the  elements  on  one  side  of  the  secondary  diagonal  be 
all  zero,  the  determinant  will  be  equal  to  the  product  of  the 
secondary  diagonal  elements  multiplied  by  (—  l)5"("-i>,  n 
beiyig  the  order  of  the  determinant. 


Examples. 


a  b  c 
0  d  e 
0  0/ 


d  e 

0/ 


adf. 


00  0  a 

=  —a 

Q  0  b 

=  -ab 

0  d 

00  h  c 

0  d  e 

a  h 

Od  ef 

g  h  k 

ah  hi 

■  abdg. 


DETERMINANTS. 


327 


Cor.  4.  The  order  of  a  determinant  may  he  raised  with- 
out altering  its  value  by  prefixing  a  column  of  zeros,  and 
superposing  a  roiu  of  elements,  the  first  of  which  raust  he 
unity,  hut  the  others  may  he  any  finite  quantities  whatever. 


ai  —  X     hi- 


EXAMPLE. 


1  X 

0     ai  —  X 
0     ao  —  X 


^2  — y 


Theorem  IV.  If  each  element  of  a  row  of  a  determi- 
nant consist  of  two  terms,  the  determinant  may  he  resolved 
into  the  sum  of  two  determinants,  the  first  of  which  is  got 
from  the  original  determinant  hy  striking  out  one  term  of 
each  of  the  elements  in  question,  and  the  second,  hy  restoring 
these  and  striking  out  the  others. 

Conversely  :  The  sum  of  any  numher  of  determinants 
which  are  alike,  except  as  regards  the  nith  row  in  each,  is 
equal  to  a  determinant  which  is  like  the  given  determinants 
except  that  each  element  of  its  mth  row  is  equal  to  the  sum 
of  the  corresponding  elements  of  all  the  given  determinants. 

Examples. 


a-{-x  d  g 

= 

a  d  g 

+ 

X     d  g 

h—y  e    h 

h  e    h 

—  yeh 

c+z  f   k 

c  f   k 

z    f  k 

a       h       c 

d      e      f 

g  ~\-m  h  k  —  n 


a  b  c 
de  f 
g  h  h 


+ 


a  h  c 
d  e  f 
m  0  — n 


\  «i  ^-2  c^d^\  +  \aih,e^d^\  +  \  a^  h,  g^  d^  \ 
=r  |ai  ^2    (^3  +  ^3+^3)    d^\ 


328 


DETERMINANTS. 


Theorem  V.  If  each  elevient  of  any  roiv  of  a  determi- 
nant he  multiplied  {or  divided)  hy  the  same  factor^  the 
determinant  will  he  m^ultiplied  (or  divided)  hy  the  said 
factor. 

Cor.  1.  If  all  the  elements  of  any  row  he  divisihle  hy  a 
common  factor,  such  common  factor  may  he  struch  out  of 
these  elements  and  ivritten  as  a  coefficient  outside  the  hars  of 
the  resulting  determinant. 

Cor.  2.  If  the  sign  of  every  element  of  a  row  he  changed, 
the  sign  of  the  determinant  will  he  changed. 


Examples. 


Za  h  c 
Sd  e  f 
3g  h  k 


=  S 


a  h  c 
de  f 
g  h  k 


4  8 
6  9 
15  20 


-=2.3.5 


1  2  4 

12  3 

13  4 


30  X  1  =  30. 


Here  the  common  factor  2  is  struck  out  of  the  first  row 
of  elements,  3  out  of  the  second  row,  and  5  out  of  the  third 
row,  and  their  product  is  written  as  coefficient  of  the  result- 
ing determinant. 


3. 


he  1  a 
ca  1  h 
ah  1  c 


ahc 


ahc  a  c^ 
ahc  h  h"^ 
ahc  c  c^ 


1  a  a^ 
1  h  h' 


0  a  h 
—  a  0  c 
-h  ~c  0 


0     a     h 
a      0 
h  -c  0 


=  (-iy 


:(-iy 


0  -a  -h 
a  0  —  c 
h     c       0 


0  —a  h 

a     0     c 
h     c     0 


DETERMINANTS. 


329 


But  this  is  the  original  determinant,  say  A,  with  its 
columns  written  as  rows  ; 

,\A  =  (-iyA',  .-.  A=:0. 

Theorem  VI.  Ani/  determinant  can  always  he  trans- 
jormed  into  a  determinant  of  the  same  order  in  which  the 
non-zero  elements  of  any  one  row  or  07ie  column  are  all  unity. 


Examples. 


«1 

^1    Ci 

^2 

h     C2 

«3 

bz   C2 

1.    Let 


Multiply  each  element  of  the  first  column  by  hiCi,  each 
element  of  the  second  column  by  CiGi,  and  each  element  of 
the  third  column  by  aibi ; 


.\a,'b,%'A-- 


aibiCi  h^c^ai 
ajJxGx  b^c^ai 
chbiCi     hc^a^ 

1 
a^biCi 
a^biCi 


=  aibiCi 


Ciaibi 
Czaibi 
c^aibi 

1 

b.2Ciai 
bsCiai 


1 

c^aibi 
CsCiibi 


=  aJ)iCiA\  say ; 


.-.A: 


aihiCi 


2.    Reduce 


3 

-5 

6 

4 


-2  7  4 
4  3  7 
3  5-2 
6-3      5 


to  an  equivalent  determinant  having  the  elements  of  its 
second  column  all  unity. 

The  least  common  multiple  of  the  elements  of  the  second 
column  is  12,  and  the  quotients  of  12  by  these  elements  are 
—  6,  3,  4,  and  2,  respectively.     Multiply  the  first  row  by 


330 


DETERMINANTS. 


—  6,  the  second  by  3,  the  third  by  4,  and  the  fourth  by  2, 
and  divide  the  determinant  by  — 6x3x4x2:=  —  144. 
The  result  is 

J^  I  -  18  1  -  42  -  24 

12   -15  1         9       21 

24  1       20-8 

8  1-6       10 


1 

-18  12 

-42  - 

-24 

144 

-15  12 

9 

21 

24  12 

20 

-8 

8  12 

-6 

10 

Ex.  74. 

Expand  the  following  determinants : 


4. 


a 

h 

c 

d  0 

e 

9 

0  h\ 

1 

0 

a' 

1 

0 

V" 

1 

c 

e 

a   b 

c 

d  e 

0 

9  h 

0 

5  0  41 

3  7 

3 

4  0 

5 

6. 


^1  yi  ^1 

0       0       02 

^'3  ya  ^3 

5       0    4 

0-2    0 

4      0 

B| 

a, 
0 

ftj  as  a, 
b^  0    h 

0 
0 

C2   Cj    c, 

e,  0    0 

X 

y 

0  0  1 

0 

X 

y 

0 

0  0 

X 

V 

y 

0  0 

X 

a 

0  e 

0 

X 

b 

Of 

X 

k 

c 

0  X 

0 

0 

d 

X  g 

h 

I 

X 

0  0  0 

0 

10. 


«1 

hi  Ci  di  Ci 

0 

h^  C2  d^  62 

0 

hO    C?3  0 

0 

^40    d,0 

0 

h  0    (4  ^5 

Show  that : 
11. 


12. 


ai  ^1  0    0 

= 

ai  5i 

X 

^1  yi 

^2  h,  0    0 

a^  h 

0C2  yi 

as  h  ^1  yi 

^4  ^4  ^2  2/2 

X         f  ^  ^ 

-\     X  y^  0 

0  -1  :r  y' 

0       0  -1  ^ 


a:      y       0     0 

-y     X      y     0 

0     -y     rr     y 

-y  ^ 


0      0 


DETERMINANTS. 


331 


Resolve  the  following    determinants    into    determinants 
with  monomial  elements : 


13. 


Cli    ^1+1      Ci'l 

a^  h.2  +  x    ^2 

«3     ^3  +  ^     ^3  1 


14. 


a-i  hi  Ci  +  x^ 

«2     ^2     ^2  +  ^^ 
«3     ^3     C^  +  X 


15. 


cci  +  x  +  y  hi  Ci 
ih--x-\-y  h^  c^ 
ch  +  x  —  y  ^3  (?3 


16. 


ai+x  hi  +  y  Ci 
a^  —  X  h^'\-y  <?2 
a^-\-x  h^  —  y  <?3 


17. 


Xi  —  ayia^  —  Zi 
X2~h  2/2  ^^  —  2^2 

^3     <^  y^  ^      2J3 


18. 


07  +  a      d 

h     y  +  e 

c  f      z  +  h 


9 
h 


Combine  into  a  single  determinant : 


19. 

Xi  3/1  z^ 
X.,  0    2:2 

+ 

^3  3/3  23 

t62    t^2    0 

^^3  y-s  2:3 

Xi  3/1  Zi  I 

20. 

cii  hi  Ci 
a,  h.,  c, 
as  h-s  (?3 

+ 

«!  as  a^ 

^1    ^3    ^4 
Ci    C,     C4 

21. 

^i  —  ai  7ji  zi 

+  yi  2 

1  «i- 

-Wl 

X.,  —  a-2  3/2  z^ 

3/2  2 

2     <^'2- 

-W2 

^3  —  «3  : 

A  2; 

i 

i2/3  2 

3     «3- 

-^3 

22.    \m-{-n~j-p     7n-{-n— p\-\-\  x  +  y  +  z      x  —  y-\~z\ 
\  X  —  y  —  z      X  4-y  —  z  \      \in  —  n  —p    m  —  n-\-p  I 


23. 


3  3/1  2^1 

+ 

'^  yi  2i 

— 

4  y2  2:2 

6  3/2  2:2 

;  5  3/3  2:3 

5  3/3  23 

10      3/i  21 

J.  \j  Xi   X2   x^ 

10      ys  Z3 
10  y,  z, 

■  J.  vj  ii7j     3/2  ^2       2-2  »^3 

5  ya  23 


.332 


DETERMINANTS. 


Show  tliat : 


24. 


26. 


26. 


c +x d+y 


a  —  u  h  - 
c  —X  cl- 


ai  Z>i  Cl 

= 

0  ^1^1 

CL'l   ^2   ^2 

(22    0     ^2 

^3  ^3   ^3 

aa^sO 

+  ai 


■y 

^2    <?2| 
^3   ^3 


=  2 


a  Z>  I  +  2  1 16  -y 
c  (i  I         I  ^  y 


0   Cl 


+  C3|0     J, 

la,  0 


ai  5, 

^1     di 

= 

(22     ^2 

c,  d. 

«3     ^3 

c,  d. 

(24    Z)^ 

c,  d. 

0    hi  Cl  di 

+  (2i 

^2     <?2     <^2 

(22     0       (?2     ^2 

^3     ^3     ^3 

as    ^3    0      (^3 

Z>4     ^4     C^4 

a^  h^  (?4  0 

+i. 


0    Cl  di 

^'i    <?3     <^3 
a4    (?4    (^4 


+  ^3 


0    bi  di 

+  ^4 

0    li  Cl 

a^  0    c?2 

(22     0      C2 

(24    Z)4     C?4 

(23     ^3    0 

III.   Evaluation. 

§  60.  Theorem  VII.  If  tivo  rows  of  a  deterrriinant  he 
identical^  the  determinant  will  he  equal  to  zero. 

Cor.  If  the  corresponding  elements  of  two  rows  of  a  deter- 
minant have  a  constant  ratio,  the  determinant  will  he  equal 
to  zero. 


Examples. 


1. 


2. 


mxi  3/1  nxi 

=  mn 

Xi  7/1  Xi 

mx2  ^2  ^^2 

^2  ^2  ^2 

m^3  3/3  nx^ 

^3  ys  ^3 

=  0. 


Tnai  +  72^1  (2i  hi 
ma^  +  nhi^  a^i  h^ 
ma^  +  nh^  a-^  h^ 


mai  (2i  hi 

+ 

ma^  (22  ^2 

mag  (23  ^3 

nZ), 

ai 

^1 

n^2 

(22 

^2 

n^a 

aa 

h. 

=  0. 


Theorem  VIII.  TAe  value  of  a  determinant  will  7iot  he 
altered  if  to  the  elements  of  any  row  there  he  added  equimul- 
tiples of  the  corresponding  elements  of  any  other  row. 


DETERMINANTS. 


333 


1. 


Ui  hi  Ci 

^2     h     (?2 
^3     ^3     ^3 


2.    Evaluate 


Examples. 


«!    bi    Cy     +     772i>i  bi    Ci 

a.i     Z>2     ^2  ^^^2  ^>>2     <^2 

a3  ^3  c^  mb^  b^  c^ 

ai  +  mbi  +  ?z<?i  Z)i  Ci 

a2  +  m&2  +  ^<?2     ^2  ^2 

as +  '^^3 +  ^^3    h  <?3 


12     3  4 

8     7     6  5 

1     3     6  10 

36  28  21  15 


nci  bi  Ci 

'^^2     ^2     ^2 

nc^  bs  Cs 


From  the  elements  of  the  fourth  column  subtract  the 
corresponding  elements  of  the  third  column,  and  write  the 
remainders  as  the  corresponding  elements  of  a  new  fourth 
column.  Do  the  same  with  the  second  column  instead  of 
the  third,  and  the  third  instead  of  the  fourth,  and  then 
with  the  first  and  second  columns  instead  of  the  third  and 
fourth.     The  resulting  determinant  will  be  : 


1 

1 

1 

1 

8 

-1  ■ 

-1 

-1 

1 

2 

3 

4 

36 

-8  - 

-7 

-6 

In  this  determinant,  add  the  elements  of  the  first  row  to 
the  corresponding  elements  of  the  second,  and  also  ten 
times  the  elements  of  the  first  row  to  the  corresponding 
elements  of  the  fourth  row  ;  the  result  will  be  : 

0 


Hence  the  given 


1111 

=  -9 

1  1  1 

9  0  0  0 

2  3  4 

12  3  4 

2  3  4 

46  2  3  4 

by  Theor.  III.,  Cor.  2,  and  Theor.  VII. 
determinant  is  equal  to  zero. 


334 


DETERMINANTS. 


3.    Evaluate                    12  4  8 

2  6  7  10 

5  9  3  1 

7  3  4  9 

Take  twice  the  third  column  from  the  fourth  for  a  new 
fourth,  t^vice  the  second  from  the  third  for  a  new  third, 
and  twice  the  first  from  the  second  for  a  new  second  ;  the 
result  is  : 


1 

0 

0      0 

= 

2 

-5 

-4 

2 

2 

-5  -4 

-1 

-15 

-5 

5 

-1 

-15-5 

-11 

—  2 

1 

7  - 

-11 

-2       1 

To  the  second  row  add  the  third,  and  from  the  result 
subtract  the  first  row,  and  write  the  remainders  as  a  new 
second  row.  To  the  first  row  add  four  times  the  third,  and 
write  the  sums  as  a  new  first  row.     The  result  is : 


-42- 

-13  0 

z= 

-14- 

-12  0 

-11 

-21 

-42- 
-14 


- 13  1  -  (- 

-12 


-)'14  i 


3  131: 

1  12 


14|2    1 
11  12 


-:  14(24-1) -322. 


1  1  1 

z=: 

a   b    c 

a'  V  (? 

0 

b  —  a 
b'-a' 


0 


—  (b  ~  a)(c  —  a) 


0 
1 


0 
1 


a^  5  +  a  c-\-a 
=  {b-  a){c  -  a)[{c  +  a)-{b  +  a)] 
=  (a  —  b)(b  —  c)(c  —  a). 

This  determinant  may  also  be  evaluated  thus : 

The  determinant  vanishes  for  a  =  ^  ;  therefore  a  —  b  is  a, 

factor  of  it.    By  symmetry,  b  —  c  and  c  —  a  are  also  factors. 

Now  the  determinant  is  of  the  third  degree  ;   there  are, 

therefore,   no  other  literal  factors   than  these  three ;    the 


I 


DETERMINANTS. 


335 


determinant  therefore  =^7n(a  —  h)(h  —  c){c  —  a),  wherein  on 
is  numerical.  To  determine  in  :  The  principal  diagonal  is 
hc\  and  the  factors  give  mhc^,  hence  ??2  =  +  l,  and  there- 
fore the  determinant  is  equal  to  {a  —  h)(h  —  c)(c  —  a),  as 
was  otherwise  already  proved. 

5.              0  X  1/  z  =    x-\-yAr'^  x  y  z 

^^   X  0  z  y  x  +  y  +  zQzy 

y  z  0  X  x-^y  -\-  z  z  0  X 

z  y  X  0  X'j-y-\~zyxO 

which  shows  that  :r  -f  3/  +  2;  is  a  factor. 

Multiply  the  first  and  fourth  columns,  and  the  second 
and  third  rows,  each  by  —  1,  which  is  equivalent  to  multi- 
plying the  determinant  by  (—  1)*  =  1 ; 

0       X  y  —  z 

X      0  —  z  y 

y  —  z  0  X 

-z     y  X  0 

and  taking  the  sum  of  the  columns,  as  before,  for  a  new  first 
column,  x^y—z  is  seen  to  be  a  factor.  Similarly,  x  —  y-{-z 
and  —x-\'y-\-z  may  be  shown  to  be  factors.  The  deter- 
minant is  of  the  fourth  degree,  and  four  linear  factors  have 
been  found; 

.".  t^^m{x-\-y-\-z){yAr^-^){^-\-^-y){^-^y  —  '^)' 

The  secondary  diagonal  is  +2^  and  the  factors  give 
—  m2;^  in  —  —1. 

:.b.  =  ~{x-\-yArz){y-\-z-x){z^rX--y){x^ry-^)' 


6. 


a?  +  1      ah 

ae         ad 

ah      h'+\ 

he         hd 

ac          he 

c'  +  1      ed 

ad         hd 

cd      d^-\-\ 

■  A,  say. 


336 


DETERMINANTS. 


Multiply  the  first  column  by  a,  and  then  strike  out  of 
the  first  row  the  common  factor  a ;  this  will  not  change  the 
value  of  the  determinant,  which  denote  by  A. 

•.  A=a'  +  1  h           c  d 

a^h  h'  +  l      he  bd 

a^c  he  c^+l  cd 

o?d  hd        ed  d'+l 

Similarly,  operate  with  h  on  the  second  column  and  sec- 
ond row,  with  c  on  the  third  column  and  third  row,  and 
with  d  on  the  fourth  column  and  fourth  row ;  then 

A=   a'  +  l      h''  c^  d^ 

a'  h''  +  l      c^  d? 

a"-  y  c'-\-\      d^ 

a'  h"  e"  c?^+l 

Take  the  sum  of  the  columns  for  a  new  first  column,  and 
write  the  common  factor  outside  the  bars. 


.•.A  =  (a^  +  Z)^  +  c^+^^  +  l) 


Subtract  the  first  row  from  each  of  the  others  ; 


1 

V 

e 

<jp 

1 

v+\ 

<? 

cP 

1 

v 

c^+l 

d' 

1 

If 

(? 

d'+l 

1  p 

c' 

d' 

0  1 

0 

0 

0  0 

1 

0 

0  0 

0 

1 

^  =  (a'  +  h'  +  e^  +  d'+l) 


=  a'  +  h'  +  e'  +  d'  +  l.   (SeeTheor.  III.,Cor3.) 

7.  X  y  z    =   X -j-y -\-z  y  z 

A—    z   X  y  x-\-y -\-z  X  y 

y  z   X  x-\-y -\-z  z   X 

.'.  X  +  ?/  +  2;  is  a  factor  of  A. 

Let  (o^  +  CO  +  1  —  0,  and  /.u)^  =  1.     Multiply  the  second 
column  by  w  and  the  third  by  uy^,  the  second  row  by  w^  and 


I 


DETERMINANTS. 


337 


X 

My   (ii^z 

■= 

i^'z 

X     lay 

i^y 

lii^Z     X 

the  third  by  w,  which  is  equivalent  to  multiplying  A  by  a>^, 
which  =  1. 

x-[~  (ay  -{-  ii^z  (ay  oi^z 
x-{-  lay  -\-  o)^z  X  coy 
X  +  o)y  +  (joi^z  iii^z    X 

/ .  X  -\-  lay  -\-  (a^z  is  a  factor  of  A. 

Operate  with  w^  instead  of  co,  and  therefore  with  w  in- 
stead of  co^,  and  X  +  co^y  +  idz  will  be  seen  to  be  a  factor 
of  A. 

.*.  ^  =^  m  {x  -\-  y  -\-  z)  {x -\-  (oy  -{-  oy^z)  (x  +  la^y  +  loz) 

in  which  771  is  numerical.     The  principal  diagonal  of  A  is 
x^,  and  the  factors  give  77ix^, 

.'.  m  —  +  1. 

.-.A  ^-(x  +  y  +  z)(x  +  o}y  +  lah) (x  +  oy'y  +  (oz) 

=  x^ -{- y^ -j~  z^  —  Zxyz. 


Ex.  75. 


Evaluate  : 


3. 


5. 


1     3-5-8 

4     7      2-6 

3  10     12      6 

-9     1     13     19 

2. 

-1-1 
1  -1 
1        1  - 
1        1 

1 
1 
1 
1 

1 

1 

1 

-1 

-5        9        5-5 
9-15       19      23 
5       19-10-15 

-  5  -  23       15-25 

4. 

1  14  15     4 
8  11  10     5 

12  7     6     9 

13  2     3  16 

17  24     1     8  15 

23     5     7  14  16 

4     6  13  20  22 

10  12  19  21     3 

11  18  25     2     9 

6. 

1111 
a    h    c    d 
a'  h'  c'  d' 
a'  ¥  c'  d' 

338 


DETERMINANTS. 


1111 

abed 
a'  b'  &  d'' 
a'  b'  c'  d' 


1111 
abed 
a'  F  c'  d' 
a'  b'  e'  d' 


11. 


1 

1 

1 

1 

a' 

b' 

c' 

d' 

a' 

b' 

(? 

d' 

a* 

b* 

c* 

d* 

(a+bf       e'  e' 

a'       (b+ef       a' 
b'  b'      (e+ay 


10. 


12. 


0  111 

1  0    a'  b' 
1  a'  0    e' 
1  b'  c'  0 

by-\-ez      bx 
ay      cz-j-ax 
az          bz 

ex 

ey 

ax+by 

13. 


a 

b  c 

0 

0 

a  b 

c 

a' 

V  c' 

0 

0 

a'b' 

c< 

14. 


X 

a 

b 

e 

c 

X 

a 

b 

b 

e 

X 

a 

a 

b 

e 

X 

15. 


X 

a 

b 

e 

d 

d 

X 

a 

b 

c 

e 

d 

X 

a 

b 

b 

e 

d 

X 

a 

a 

b 

e 

d 

X 

16. 


X  y  y  y  y 

y  X  y  y  y 

y  y  X  y  y 

y  y.y  ^  y 

y  y  y  y  X 

17. 


X 

X 

y 

X 

y  y  y 

X 

y 

X 

X 

X 

y 

X 

y  y\ 

18. 


{b+cy 

b-" 


e" 

{c+df 

a' 


a} 

{a+by 


19. 


a -\- b  -\-  e -\-  d 
a  —  b  —  e-\-  d 
a  —  b  -{-  c  —  d 


a  —  h  —  e-\-  d 
a -\-  b  -\-  c -{-  d 
a-{-b  —  c~  d 


a  —  b  -{-  c  ~  d 
a-\-b  —  e~  d 

a  •{- b  -\-  c -\-  d 


20. 


{b+c+dy 

a'' 


a'- 


{e  +  d^ay 


{d+a+by 


d-" 
d' 
d'' 

{a  +  b  +  ey 


DETERMINANTS.  339 


§  61.  Any  determinant  of  tlie  third  order  may  readily 
be  evaluated  by  the  following  method,  called  The  Method  of 
Sarins.     Let  the  determinant  be 


ai 

b,  Ci 

a^ 

b,  c. 

% 

b^  Cs 

Repeat  in  order  the  first  and  second  rows  below  the 
determinant  (or  the  first  and  second  columns  to  the  left  of 
it)  ;  thus, 

ai     bi     Ci     or,     ai     bi     c^     a^     b^ 

\     /  \,X    X     /, 

a^     62     (?2  «2     ^2     G'l     ^2     ^2 

X,  X  /.X    X    ^z. 

^3        ^3        Cz  «3        ^3        <?3        <^3        O^ 

X7.  X 

«!         Oi         Cx 
«2         \         G2 

Form  the  product  of  the  three  elements  in  the  principal 
diagonal,  and  also  of  the  three  in  each  of  the  two  lines 
immediately  following  the  principal  diagonal,  and  parallel 
to  it.     In  this  case,  these  products  are  : 

ai^2<?3,  <^2^3^i,  «3^i<?2 ;  (or  a^^c^,  b^c^a^,  c^a^b^). 
Next,  form  the  product  of  the  three  elements  in   the 
secondary  diagonal,  and  also  of  the  three  in  each  of  the 
two  lines  immediately  following  the  secondary  diagonal, 
and  parallel  to  it.     In  this  case,  these  products  are  : 

aJ)2Ci,  ajb^c^,  aJ)^C2 ;  (or  Cib^a^,  aicj}^,  b^a^c^). 
From  the  sum  of  the  former  three  products  subtract  the 
sum  of  the  latter  three  ;  giving  in  this  case  : 

<^i^2^3  +  ct'ihci  +  ct'db^Ci  —  {ajy^c^  +  a^b^c-^  +  aj)^c^, 
or,  taking  the  products  derived  from  the  right-hand  ar- 
rangement as  given  above  : 

«1^2^3  +  ^1^2«3  +  ^1<^2^3  —  (<?1^2«3  +  «1<?2^3  +  b^a^C^). 

The  given  determinant  is  equal  to  either  of  these  expres- 


340 


DETERMINANTS. 


sions,  whicli  are  of  the  same  value,  as  may  easily  be  seen, 
for  they  differ  merely  in  the  order  of  their  terms,  and  in 
the  order  of  the  factors  of  those  terms. 

In  practice,  it  will  soon  be  found  sufficient  merely  to 
imagine  the  rows  (or  the  columns)  repeated. 

§  62.  The  following  theorem,  which  is  an  immediate 
consequence  of  Theor.  VIII.  and  Cor.  2,  Theor.  III.,  often 
affords  the  quickest  and  readiest  means  of  evaluating  a 
detei-minant  with  numerical  elements. 

So  arrange  the  given  determinant  that  none  of  the  elements 
bjn,  Cm,  .  .  .  km  shall  he  zero,  and  thai  all  the  elements  of  the 
mth  row  after  \^  shall  he  zero,  then 


•Si 

S2' 


«!  by  Ci  . 

.k... 

tti  5j  C2 . 

.  4 . . . 

a-i  h  c,. 

.  4 . . . 

Onfim.  •    •   •   1^11 


h, 
hm 

h, 

hr„. 


Ctm        h^ 


m-j-1  ^m+1  I 


If  m  =  1,  the  elements  of  the  first  row  will  be  : 


cii  hi 
a^  &2 


^2    Co 


ki  ti 


Vi^ 


This  method  of  evaluation  is  known  as  Condensation, 


DETERMINANTS. 


341 


Examples. 


1  2  3 

-6 

4  1  2 

0 

3  0  1 

4 

0  2  1 

1 

Evaluate : 


Here,  since  none  of  the  inner  elements  of  the  first  row  is 
zero,  we  may  take  77^  =  1 ;  then,  operating  on  the  first  two 
rows,  we  mentally  evaluate 


1  21 
4  1 


2  31 
1  2 


and  write  the  results,  which  are  —  7,  1,  12,  for  the  first  row 
of  the  new  determinant ;  similarly,  we  proceed  with  the 
first  and  third  rows,  and  then  with  the  first  and  fourth 
rows.  This  gives  a  determinant  of  the  third  order,  which 
we  divide  by  6,  the  product  of  2  and  3,  the  inner  elements 
in  the  first  row  of  A,  and  we  thus  obtain 


A  =  i 


This  determinant  may  be  evaluated  by  the  Method  of 
Sarrus,  or  the  condensation  may  be  repeated.  Condensa- 
tion gives  *" 

-4  -1 


1 

12 

-7 

1 

4 

-6 

9 

18 

=: 

-3 

1 

3 

2  - 

-4 

9 

2 

-4 

3 

26     19 


76 +  26 --50. 


Ex.  76. 

Evaluate  by  the  Method  of  Sarrus  : 
1. 


1  1  1 

2. 

1  2  3 

3. 

2 

-1 

1 

12  4 

2  3  1 

1 

2  - 

-1 

13  9 

3  1  2 

-1 

1 

2 

342 


DETERMINANTS. 


4. 


7  6  7 
1  -2  1 
3       1  -2 


0  17 
21  0 
13-2 


Evaluate  by  condensation : 

1-1  2  0 
12  0  3 
2  0  3  1 
0      3-14 


8. 

0  118 

■ 

5  0  11 

4  10  1 

3  110 

1 

1 

1 

2 

3 

4 

1 

1 

1 

3 

4 

5 

1 

1 

1 

5 

6 

7 

9. 


•2001 
1  -3  -1  -2 
3  1-70 
0      3       1-5 


11. 


10. 


1 

1 

0 

-1 

2 

2 

1 

0 

3 

2 

3 

1 

5 

4 

2 

3 

0  111-1 
-10  111 
1-10  11 
11-10  1 
111-10 


IV.  Multiplication. 

§  63.  Theorem  IX.  The  product  of  two  determinants  Ai 
and  A2  of  the  same  order,  is  a  determinant  such  that  the 
element  in  its  ipth  row  and  qth  column  is  the  sum  of  the 
products  of  the  elements  of  the  ipth  row  of  Aj  each  multiplied 
into  the  corresponding  element  of  the  qth  column  of  A2. 

Writing  a^^,  x^^,  and  A^^,  for  the  (p,  q)ih.  element  of  Ai, 
A2,  and  of  their  product  respectively,  then 


i>?  i>i 


^\,  q     \     ^p,  2  ^2,  q     \     ^p,  3  ^S,p     \     ^tC. 


Before  forming  the  product  as  above,  Aj  or  A2  may  either 
or  both  of  them  be  transformed  by  rearranging  the  rows  or 
the  columns,  or  by  changing  rows  into  columns.  The  pro- 
duct of  the  same  two  determinants  will  therefore  appear 


DETERMINANTS. 


343 


under  different  forms  depending  on  the  arrangement  of  its 
factor-determinants,  but  these  forms  will  all  have  the  same 
value.  If  one  of  the  determinants  to  be  multiplied  together 
be  of  a  lower  order  than  the  other,  its  order  must  be  raised 
to  that  of  the  other.     (Cor.  4,  Theor.  III.) 


Examples. 


«2     ^2   I  I   ^\     3/2 


tti   bi 

X 

Xi    X2 

= 

(22     ^2 

yi  y-i 

«!  a.2    X 

Xi     X.2 

yi  y2 

=^ 

X 

^1  yi 
^2  y2 

^^^ 

CtiXi  ~y~   0^X2^ 
^2*^1     l~   ^2*^2) 

«i^i  +  ^lyi, 
«2^^  +  ^2yi, 

«i^i  +  «2yi, 
^1^1  +  ^2yi, 

biXi  -\-  1)2X2, 


cciVi  +  ^iy2 
«2yi  +  ^2y2 

«i^2  +  ^iy2 

«2^2  +  ^2y2 
a^X2  +  a2y2 
^l^'2  +  ^2y2 

chVi  +  ct2y2 

%i+%2 


ai  ^1  Ci 

X 

X^  X2  x^ 

^2     ^2     <?2 
aa    Z>3    <?3 

Vi  yi  y3 

Zx    Z2    Z; 

«i^i  +  ^lyi  +  ^i2;i   aiX2  +  ^i?/2  +  ^i2;2   «i^3  +  ^ly^  +  ^12:3 
«2^i  +  ^2yi  +  ^22^1   «2^^2  +  ^2y2  +  ^2^2   ci.2x.i  +  ^2y3  +  <?22;3 

a^oci  +  ^syi  +  ^32:1    a-iX2  +  b.,ij2  +  c.^Z2    a^x-^  +  b^y^  +  ^32:3 


cCi  bi  Ci  di 

X    ^1  ^2 

^ 

a2  ^2  C2  d2 
a-i  ^3  c-i  d^ 
a^  ^4  (?4  (^4 

yi  y^ 

«!  Z>i  ^1  di 

a2  &2  <?2  C^2 

as  Z)3  <:3  C?3 

a^  b^  Ci  d^ 


X 


1^7  J    X2    X^    X^ 

yi  y2  y3  y* 

0    0     1     ;24 

0001 


<^i''^i+^iyi  ctiX'2+biy2  CLiX-i+b^y^+Ci  a^x^+biiji+c^z^-^d^ 

a2Xy+b2yi  (uxf\-b,y2  a2^34-^2y3+C2  a2-'^4+^2y4+^^22^4+^4 

«3-^^i+^3yi  (i^xA-b^y2  (^^^x-^+b^y-i+c-i  chXi-\-b.iyi+c.iZi+d-^ 

a^x^^b^y^  ChX2+biy2  a^x-i+biyor^-Ci  a^Xi+biyi+c^Zi+di 


Here  x^,  y^,  x^,  3/4,  z^  are  wholly  arbitrary,  and  may  be 
made  all  zero. 


344 


DETERMINANTS. 


4.    I  ay  Z),  C-,  d^  ?  =^ 


ai  hi  Ci  di 

X 

a<i  ^2  <^2  d^i 

<23     Z>3     <?3     C?3 

a^  h^  c^  d^ 

di 


h 

h 

Z', 

tti  - 

-a, 

-a. 

d. 

C?3 

J, 

C2    - 

-<73 

-C4 

0  \aih^\-\-\cid^i\  l^i^sl+ki^al  |^i^4|+i^ic?4| 

«2  ^1  l  +  l  ^2  C?1  I  0  I  ^2  ^3  1+1  ^2  <^3  1    I  «2  ^4  |  +  |  ^2  ^^4! 

<3^3  ^>>1  l+l  Czdx\    I  ^3  Z>2  l  +  l  C3  C?2  I   I  0  I  a3  ^4  l+l  ^3  (^41 

K^ll  +  k^C^il    |a4^2|  +  k4<^2|   l^l^sl+k^C^sl  0 

This  is  a  skew  symmetrical  determinant  for 

(I  ai  Z^2 1  +  I  ci  d,  I)  +  (I  ^2  ^1 1  +  I  ^2  ^1 1)  =  0, 

by  Theor.  II. ;  and  the  same  holds  for  every  other  pair  of 
conjugates. 

§  64.  If  from  A,  a  determinant  of  order  n,  there  be 
erased  m  rows  and  m  columns,  the  determinant  formed 
from  the  remaining  rows  and  columns  taken  in  order,  is 
called  a  Minor  of  A  of  order  m  —  n.  The  minors  obtained 
by  erasing  one  row  and  one  column  of  any  determinant  are 
called  the  Principal  Minors  of  that  determinant. 

Two  minors  which  are  so  related  that  the  rows  and  col- 
umns erased  in  forming  one  of  them  are  exactly  those  not 
erased  in  forming  the  other,  are  called  Complementary  Minors. 

Thus,  I  ai  &2  <?3 1,     I  <^2  ^3  d^  |,     |  ^i  ^4  ^5 1 

are  third-order  minors  of  |  a^  \  c^  d^  <?5 1,  and  their  comple- 
mentaries  are  the  second-order  minors, 

1^4^5  1,        kl^5|,        |0^2<^3|, 

respectively. 

I  ^1  ^4 1  and  I  (?2  c?5 1        . 

are  seco»d-order  minors  of  |  a^  h^  Cj,  d^  ^s/e  |,  and  their  com- 
plementaries  are  the  fourth-order  minors. 


respectively, 


I  «2  ^3  c^s/e  I  and  I  ai  h^  e^f^  j 


DETERMINANTS.  345 


The  principal  minors  of  |  cii  b^  c^  di^  e^  \  are 

I  ^2  <?3  d^  65 1,     I  ^1  <?3  <^4  ^5 1,     \bi  c^d^e^l  ,,, 

I  <^2   <?3   <^4   -^S  I,         I  Ctl   <?3   <^^4   65    |,     .    .   . 

complementary  to  (2i,  a2,  a^,  .  .  .  ^i,  h^,  .  .  .,  respectively. 

Hence,  if  in  any  determinant,  A^^ ,  denote  the  comple- 
ment of  ap^q  (page  000),  (— 1)^+^^_p,^  will  be  the  principal 
minor  complementary  to  a^^^. 

Theorem  X.  If  any  m  rows  of  a  determinant  he  selected, 
and  every  possible  Tninor  of  the  rath  order  be  formed  from 
them,  and  if  each  be  Tnultiplied  by  its  confiplementary  and 
the  product  affected  with  -\-  or  —,  according  as  the  sum.  of 
the  numbers  indicating  the  rows  and  the  columns  from  which 
the  minor  is  formed  be  even  or  odd,  the  sum  of  these  products 
will  be  equal  to  the  original  determinant. 

Thus,  the  first  two  rows  of  |  ai  b^  c^,  d^  \  give  the  six 
minors, 

I  (^x  K  |,     I  «i  (?2  |,     I  ai  c?2  |,     I  ^1  6?2  I,     I  61  d^  I,     I  ^1  J2  I, 

whose  complementaries  are 

I  ^3  d^  I,     I  ^3  d^  I,     I  b^  c^  |,     I  a^  d^  |,     |  a^  c^  |,     |  a^  b^  |, 

and  the  sums  of  the  numbers  indicating  the  rows  and  the 
columns  from  which  the  first  six  minors  are  formed,  are 

(1  +  2  +  1  +  2),     (1  +  2+1  +  3),     (1  +  2+1  +  4), 
(1  +  2  +  2  +  3),     (1  +  2  +  2  +  4),     (1  +  2  +  3  +  4), 

.*.  \  aih,c^d^\  =  \  «!  ^2  I    I  ^3  <^4  I  —  I  «!  ^2  I    I  h  d^  I 

+    I    «!    c/2    I        I    ^3    ^4    I    +    !    ^1    <?2    I        I    «3   <^4    i 
—    \hdi\       I    ^3    C4    I    +    1    Ci    (^2    I       I    «3   ^4    I 


346 


DETERMINANTS. 


Similarly,  the  second,  third,  and    fifth    columns    being 
those  selected, 

'  I  ai  1)2  Csd^e^l  =  \  bi  c^  e^\  \  a^  dr,\  —  \  h^  c^  e^ 

+    I  Z>1   ^2   ^5    I  I    «3   C?4    I    +    I    ^1   ^3   e^ 

—  I  ^1    ^3   ^5    I  I    <^2   C?4    I    +    I    ^1    ^4   ^5 

—  I  ^2  ^3  <?4  I  I  ai  c^5  I  +  I  h.,  Cs  ^5 
~  \  hc^e^l  I  «!  (is  I  +  I  ^3  ^4  ^5 


1       1    «3 

^ 

i       l«. 

C?5 

1       1    «2 

^3 

1       1    «1 

C^, 

1       l«l 

d. 

Ex.  77. 

Perform   the    following   multiplications,   expressing  the 
results  in  determinant  form  : 


4. 


6. 


(7-2     ^2 

2  5 

3  6 


^1  X2 

-3  - 

9. 


\S  5\\x  u 
\  4:  6\\y  V 


2  5 

3  4 


8. 


2-3 


1—9 

"2        ^ 

i  -3 


1 

2 

3 

7. 

221 

2     6     8 

3    1^    3^ 

■3     7     9 

2 

4 

K 

6     3     2 

2     3     4 

3 

4 

-7 

T    7     7 

Hi 

3     7     9 
if    i 

A      5-y 


A       a  +  y 


3  5 

4  2 


10. 


a  — y      A  (7 

9       f    <^~y 


a  +  y      h         g 
A       h  +  y      f 

9       f    <^  +  y 


lad" 

1  b   b' 

Ice' 

a'      b'      c' 

—  a  —b  —  c 

1        1       1 


11.      X   yi 

yi   X 

wherein  2^  — —  1. 


y  2:^ 

\zi  y 


Z     XI 

xi  z 


DETERMINANTS. 


347 


18. 


19. 


20. 


21. 


12. 

a -{-hi     ~  c  ^  di        X  -\-yi       u-j-  vi 
c  -\-  di         a  — hi      —u-\-vi     x  ~  yi 

13. 

a  -{-  h      a         h 
c       h-\-c      h 
c         a      c-\- 

a  +  h  +  lc       --la           ~ih 
~ic       b  +  c+ia       ~ih 
a          ~ic            —2a       c+a+i^ 

14. 

a  a  a  a 
a  h  h  h 
a  h  c  c 
a  h  c  d 

1 
-1 
0  - 
0 

0  0     0 

1  0     0 
-1       1     0 

0  -1     1 

15. 

X  y  z 
z  X  y 
y  z   X 

x^  z'  y' 
y'  x'  z' 
z'  y'  x' 

16. 

1    —X       X 

a       he 
h        c     d 
c        d     e 

'  -x^ 
d 

e 

f 

;i     :r     0 
0     1^^ 
0     0     1 
0     0     0 

0 

0 

X 

1 

17.      «!  +  (22     ^2  +  a^i     a^  +  «! 
hi  +h.,     h2  +  h.,     h^  +  b^ 


1^ 

-2 

1 

1 

1 

1- 

-2 

1 

1 

1 

1- 

_2 

2 

1 

1 

1 

«l+«2+«3  ^1+^2+^3     Cy^C^+Cz  dr\-d^+d, 

«2+<^3+«4  ^2+^3+^4    ^2+^3+^4  d^+d-^+d, 

^3+«4+«l  ^3+^4+^1     ^3+^4+^1   d^+d^-\-d^ 

a^+ai+a^  ^4+^1+^2  <?4+^i+<?2  d^+d^+d^ 


^. 


-h. 


C2  —Oi  -c, 
^2  0  —  ^1  0 
0  0  0-1 
0     0-1        0 


«!  hi  ei  0 

0  cii  hi  Ci 

a^i  Z>2  <?2  0 

0  a.  ho  c. 


Z>2  C2  d^ 

C2  d^  0 

c/2  0  0 

0  0  0 

0  0  0 

0  0  0 


-hi 
■Ci 
-  di 

0 

0 
■1 


■Ci 

■di 
0 
0 

-1 
0 


«!  hi  Ci  di  0  0 
0  ai  hi  Ci  di  0 
0    0    cti  hi  Ci   di 


Ch 

h 

c,. 

d. 

0 

0 

0 

a, 

b. 

C; 

d. 

0 

0       0       ^2      ^2      ^2     ^2 


1 

-  2x  x' 

u^  v^  w^ 

1 

-^■I/f 

U     V     lU 

1 

-2z  z' 

1  1  1 

348 


DETEUMINANTS. 


22.      1  —3a  3a'  -a' 
1  -3b  Sb'  -b' 

\  -3d  3(P  -d' 
and  deduce  therefrom  that 
9  (a  -  by  (a  -  cy  (a  -  df  (b  -  cj  (b  - 


a' 

I' 

c' 

d? 

a' 

b' 

<? 

^ 

a 

b 

c 

d 

1 

1 

1 

1 

-d)\c-dy 


52 
54 


...    5, 

. . .    S^j^i 

...  s„+, 

•••     ^2^ 

=:[{a-by{c-dy+(b-d)\c-af+(a-d)\b-cyj. 

23.  If  s^  =  ar  +  cur  +  a.r  +  -  +  a,r.  then  will 
1    «!   a/  ...  ai""  '  =   5o    Si 

i       0^2      ^'2        .^.     ^2  5^       S^ 

1    a3   as'  ...  as**  ^2    53 

1    a^  a„    ...  a^  5^  s„_^i  5„_^2 

24.  If  A  =  I  ai  ^2  c,  ...  4  1   and  A'-  |  A,  B,  C,  ...  J^,  | 

wherein  yli,  —^2,   -^s,  •••  — -^i?   -^2,  — -^3,  etc.,  are 

the  principal  minors  of  A  ;   and  if  aj,  —a~^,  a^,  ...  —/3i, 

Pij  —  Aj  etc.,  are  the  principal  minors  of  A',  prove 

that 

AA':=A"  and  aiA  =  aiA",  a2A=:a2A**,  etc. 

Note.  The  determinant  A'  is  called  the  Reciprocal  of  the  deter- 
minant A  ;  and  the  elements  A-^^,  A^,  ...  B^,  B^,  etc.,  are  called  Inverse 
Elements  with  respect  to  %,  a^,  ...  ^j,  h.^,  etc. 

25.  Prove  that  a  minor  of  the  order  m,  formed  out  of  the 

inverse  constituents,  is  equal  to  the  complementary 
of  the  corresponding'  minor  of  the  original  deter- 
minant multiplied  by  the  (m  —  l)th  power  of  that 
determinant. 

V.    Applications. 

§  65.    To  solve  the  simultaneous  linear  equations, 

a^x  +  b^y  +  CiZ  =  di,  (1) 

a^x  +  b,,y  +  c.,z  =  ^2,  (2) 

«3^  +  hy  +  c-6^  =  4-  (3) 


DETERMINANTS. 


349 


Let  V  =  I «i^2^3 1,  Va  ="  \  ^i^2^3 1,  V6  =  I  chd^c^  \jVc  =  \ ctA^sl 
and  let  Ai,  A^,  etc.,  denote  the  complements  of  a^,  a.^,  etc., 
in  V- 

Multiply  (1)  by  A,,  (2)  by  A,,  (3)  by  A,,  and  add. 

.-.   (ai^i  +  a. A.,  +  chA^)x  +  {h^A^  +  h2A^  +  Ms)^ 
+  (^1  A,  +  ^2^2  +  c-6  A,)  z  =  (d^A,  +  d,A2  +  cI^A,). 
Therefore,  by  Theor.  III.,  Cor.  1,  and  Theor.  VII., 

By  using  B^,  B^,  B^  instead  of  A^,  A^,  A^,  we  obtain 

vy  =^  Vh, 

and  using  Q,  Q,  Q  instead  of  Ai,  A^,  ^3,  gives 

V^  =  Vc- 
The  method  here   exhibited  is  evidently  applicable  to 
the  case  of  n  linear  equations  containing  n  unknown  quan- 
tities. 

2.  Given  the  above  linear  equations  (1),  (2),  (3)  to  find 
the  value  of  ax  +  y8?/  +  yz. 

By  Theors.  VIII.  and  VII. 

ax  -\-  Py  -\-yz  a  (3  y     ^0. 

aiX  +  biy  +  CiZ  ai  b^  c^ 

a^x+h^y  +  c^z  a^  h^  c^ 

a^x  +  b^y  +  c-iZ  ag  h  <?3 

Therefore,  substituting  from  the  equations  (1),  (2),  (3), 

ax  +  py  +  yz     a      /5     y     =-0; 
di  ai    bi    Ci 

6^2  Ct'2        ^2        ^2 

ds  as    ^3    <?3 

.-.    (cot  + /gy  +  72;)  V  -  aVa  -  i^Vfe  -  yVc  =  0  ; 
•••    ax  +  py  +  yz  =  (aVa  +  ^Vft  +  yVc)  "-  V- 
This  result  necessarily  includes  the  solution  of  the  equa- 
tions found  above. 


360 


DETERMINANTS. 


3.  To  determine  the  condition  that  the  homogeneous 
linear  equations 

a^x  +  h^y  +  c^z=-0,  (1) 

^2^7  +  %  +  (?.,2;  =  0,  •     (2) 

a^x  +  h^y  +  c^z-^0,  (3) 

may  coexist  for  values  of  x,  y,  and  z  other  than  zero. 
Multiply  (1)  by  A,,  (2)  by  A,,  and  (3)  by  A„  and  add. 

.•.   \x^=0, 
and  therefore  if  x  be  not  zero, 

v  =  o. 

4.  To  find  the  condition  that 

a^x"^  +  hiX  '\-  Ci-=0  (1) 

and  a2x'^  +  b2X  +  ^2  =  0  (2) 

may  have  a  common  root. 

Multiply  each  of  the  given  equations  by  x ;  the  resulting 
equations  together  with  the  given  equations  constitute  the 
four  simultaneous  equations, 

aiO^  +  hix'^  +  C\^  '=  0, 

aix"^  +  hiX  +  Ci  =  0, 
a2X^  +  h.iX'^  -{-  c^x  =0, 

a^x^  +  h^x  +  C2^=0, 

which  are  to  be  satisfied  by  values  of  x^,  x^^  and  x  other 
than  zero ;  hence,  by  No.  3  above, 

-0. 


a, 

\ 

Ci 

0 

0 

a, 

h. 

Cx 

a^ 

h 

Ci 

0 

0 

a^ 

K 

Ct 

351 


This  is  also  the  condition  that 

aiX^  +  hiX  +  Ci  and  a^x^  +  ^2^  +  ^2 
may  have  a  common  factor. 

5.    To  find  the  condition  that 
ax^  +  3  hx"^  +  3  ca:  +  (i 
may  have  a  square  factor. 

Let  (x  —  my  be  the  square  factor.  Divide  the  given 
expression  by  ^  —  tti,  and  the  quotient  hj  x  —  7n  )  the  two 
remainders  thus  obtained  must  both  vanish.  These  re- 
mainders are 

aTYi^  +  3  hiri^  +  3  cvi  +  d 
and  am}  +  2  htn  +  <?. 

.-.   am^  +  3 Z)m^  +  3 cm  +  d=0, 
and  am.'^  +  2  Z>m  +  6?  —  0. 

Multiply  the  latter  equation  by  m,  and  subtract  the 
product  from  the  former. 

.-.   hvi^ -\-2cm.  + d=0. 

Combining  this  equation  with  the  second  of  preceding, 
the  condition  required  is  found  to  be 

that  ax"^ -\-2bx-\- c  ^=0 

and  hx^  -\-2cx  +  d=0 

shall  have  a  common  root,  and  the  condition  for  this  has 
been  found  in  No.  4  above;  viz., 

-  0. 


a  2h 

c    0 

0     a 

2b  c 

b   2c 

d    0 

0    h 

2c   d 

352 


DETERMINANTS. 


6.    To  find  the  condition  that  the  expression    • 
ax^  +  hy^  +  cz^  +  2fyz  +  2gzx  +  2  hxy 
may  be  the  product  of  two  linear  factors. 

Let  the  factors  be  a^x  +  piy  +  jiZ  and  a^x  +  ySgy  +  722;. 

Multiply  these  together,  and  equate  the  coefficients  of 
the  product  with  those  of  like  powers  of  the  variables  in 
the  given  expression. 

2/  =  Ay2  +  A71       2  ^  =  ai72  +  a27i       2  A  =-  ai/52  +  a2^i 
1 


a  h  g 
hbf 

9  J    ^ 


aia2  +  a2ai  aj^a  +  ^2^  ^i72  +  ct27i 
P,a,  +  P,a,  p,p,  +  p,P,  A72  +  A71 
7i<*2  +  72«i     71/^2  +  72^1    7172  +  7271 


71  72  0  I 


^2     A     72 

«!    ^1    7i 

0    0    0 


:0. 


Hence  the  required  condition  is  that 

:0. 


a  h  g 

h  b  f 

9  f  <^ 

7.    If 


and 


then  will 


y  =  a,X+(3,Y+y,Z  (1) 

Z--^a,X+l3.,Y+y,Z 
ax''  +  by'  +  cz'  +  2fyz  +  2gzx  +  2  A^y 
-  ^X^  +  J^Y'  +  (7^2  +  2i^F^ 

+  2GZX+2JIXY,  (2) 


A  H  G 
HBF 
O    F    C 


ai  /5i  71 
^2  ^2  72 
as  A   73 


a  h  g 

hbf 
9  f  ^ 


DETERMINANTS. 


353 


Substitute  for  x,  y,  and  z  in  (2)  their  values  in  (1),  and 
equate  coefficients  of  like  powers  of  X,  Y,  and  Z. 

.'.  A  =  aa.^  +  ha}  +  C(xi  +  2/a2a3  +  ^ga^pii  +  ^ha^a^, 
B  -  a^,'  +  h^.}  +  e^i  +  2/A/?3  +  2^ft^i  +  2  A^A 
^         (7  -  ay,^  +  ^y/  +  ^y/  +  2/y,y3  +  2^y3yi  +  2  Ay^y, 
F  -  aySiyi  +  Z^^.y^  +  ^^^373  +7(^73  +  A72) 

+  ^  ( A71  +  A73)  +  ^  ( A72  +  A71) 
6^  :=:  a7iai  +  Z^yatta  +  Cygag  +/(72a3  +  730^2) 

+  9  (ystti  +  71^3)  +  ^  (71^  +  72^1) 
H^  aa^Pi  +  5a2/S2  +  Ca-Sz  +f{a^l3,  +  as^a) 

+  9  (a,^i  +  aiA)  +  A  (a  A  +  a^ft). 


Now 


ai  A   71  ' 

^2     /^2     72 
^3     ft    73   i 

a  h  g 
hh  f 

9  f  ^ 

tti     02     03 
iSi    ft   ft 

a  h  g 
h  h  f 

«!  ^1  71 
^2  ft  72 

71     72     73 

9  I  ^ 

as  ft  73 

aai  +  Aaa  +  ^tts  Aai  +  ^a2+/a3  ^ai+Zaj  +  Cttg 
«ft+Aft+^ft  A/?i+^ft+/ft  ^^i+/ft+cft 
a7i+A72  +  (773   hy^  +  hy^+fy-^  ^7i+/y2  +  <^73 


AUG 
H  B  F 

G  F  Q 


ch  Pi  71 

^2     ^2     72 

as  ft  73 

8.    Eliminate  ^,  y,  and  2;  from  the  equations 

a;r'  +  %'  +  ^2'  +  2/y2  +  2gzx  +  2  hxy  =  0,  (1) 

^1^  +  ky  +  "^^^z  =  0,  (2) 

Jc.^x  +  4y  +  '^2^!  =  0.  (3) 


354 


DETERMINANTS. 


Fii^st  Method.    Let  X^  and  Ag  be  homogenous  linear  func- 
tions of  X,  y,  and  z,  such  that  Ai(2)  +  X2  (3)  =  (1). 

.-.  ax^  +  hy'^  +  cz^  +  2/y2;  +  2^2:0;  +  2  A^y 

=  (V  +  Ziy  +  miz)  Ai  +  (^2^^  +  4?/  +  77122;)  A2. 
.-.  ax+hij  +  gz==^  Jc.X^  +  ^2^2,  (4) 

hx  +  by  -{-  fz  =  liXi  +  ^2^2,  (5) 

^^  +  /y  +  ^^  ^  ^Ai  +  ^2^2.  (6) 

Now  eliminate  x,  3/,  2;,  Ai,  A2  from  (2),  (3),  (4),  (5),  (6)  by 
the  method  exhibited  in  No.  3,  page  350. 

a     h    g       Ici     IC2     =  0. 
h     h   J       li       4 
g    f    c       mi    m2 

^1      li     TTli     0         0 

X'2    4    ^2    0       0 

Second  Method.     Multiply   (2)   by  :r,  y,  and  2;,  succes- 
sively, and  (3)  by  x  and  by  y. 

.*.  kix'^  +  mi2;a;  +  4^y  =  0, 

4y'         +miy2;  +^ia;y=-0, 

m^z^  +   4y2;  +  ^i2;:z;  —  0, 

Jc2X^  +  m22;:x;  +  4^y  =  0, 

4y'         +m2y2;  +^2^y  =  0. 

Now  eliminate  x'^,  y^,  2;^  y2;,  2;:^,  :ry  from  these  five  equa- 
tions and  (1),  by  the  method  of  No.  3,  page  350. 

a  h  c  2/  2g  2h   =0. 

k,  0  0  0  mi  4 

0  4  0  mi  0  ki 

0  0  m,  4  h  0 

^2  0  0  0  7n2  I2 

0  I2  0  7n2  0  ^2 


DETERMINANTS.  355 


9.    To  solve  the  simultaneous  equations 

a^x"  +  2  h^xy  +  hiif  =  m^,  (1) 

a^x"^  +  2  h-^xy  +  ^23/^  =  '^2-  (2) 

Write  them  in  the  form 

(aiX  +  hi7/)x  +  (hiX  +  %)y  =  mi, 

(a2^'  +  ^23/)^'  +  (/hx  +  h{y)y^  m^. 

Let       Y 


aix  +  h^y  Kx  +  hiy 
a^x  +  h^y  h^x  +  b^y 

vy  =  —  (<^2^  +  ^2y)  '^^1  +  {chx  +  Aiy)  ^2. 

.'.  (v  +  |^i^2|)^  +  |^im2|?/  =  0,  (3) 

—  I  ai  ma  |:?;  +  (V  —  i  Ai  ??i2 1)  y  =^  0.  (4) 

/.   (v'  —  1  Ai  ma  p)  +  I  <^i  ^2 1  I  ^1  ma  I  =  0. 
.*.   V  =  ±  V(l  ^1  ^2  r  —  I  «i  ^2 1  I  ^1  ma  I). 
Hence  v  i^^y  be  treated  as  known,  and  then  by  (3), 


y- 


I  K  "^h 


Substitute  this  value  of  y  in  (1),  and  there  will  result  a 
pure  quadratic  in  x,  from  which  the  value  of  x  may  be 
immediately  obtained. 

10.    To  solve  the  simultaneous  equations 

aix""  +  bi7/  +  2  hixy  =  mj,  (1) 

«2^  +  hy  =  mj.  (2) 

This  may  be  treated  as  a  particular  case  of  the  preced- 
ing, or  otherwise  as  follows  : 


356 


DETERMINANTS. 


Let 


(3) 


Write  tlie  given  equations  in  the  form 

a^x  +  h^y  —  wi^ 

V    ==--  {cLxX  +  Ky)  ^2  —  ( Ai'o;  +  h^y)  a^. 

Vy  ==  —  «2^>^^i  +  (<^i^  +  hiy)  m2. 

( V  +  Aima)  X  +  ^i^22y  —  ^2'^^!  ==  0, 

—  aimao;         +  ( V  —  K'ith)  y  +  aami  =  0, 
a^x  +  ^22/  —  '^2     —  0. 

-0. 


V  +  Aim2        h{m2  ^2^^i 

«2  ^2  ^2 


(22  ^2  '^2 


which  pure  quadratic  gives  at  once  the  two  values  of  v, 
which  may  consequently  be  treated  as  known.  Then  from 
(2)  and  (3), 

a^x  +  ^23/  =  m2, 

(a.1^2  —  hia2)x  +  (hih^  —  hiO^  =  V ; 

two  linear  equations  from  which  to  find  x  and  y. 


11.    Eliminate  x  from  the  simultaneous  equations 

x^  —px^  +  qx  —  r  =  0,  (1) 

y  =  ai  +  h,x  +  cix\  (2) 


DETERMINANTS. 


357 


Multiply  (2)  by  x,  and  in  the  result  substitute  the  value 
of  x^  given  by  (1). 

.-.  x7/  =  c^r  +  (ai  --  c,q) x  +  (b^  +  c^p) x^ 

=  ^2  +  h^x  +  c^x"^,  say.  .  (3) 

Eepeat  with  (3)  and  (1)  instead  of  (2)  and  (1). 

.-.  x'y  -=  c.,T  +  (a.,  —  c^q)  ^  +  (^2  +  c^p)  x" 
fe  =:  ^3  +  Z^g^r  +  c^x'',  say. 

Eliminate  x  and  x^  from  (2),  (3),  and  (4). 

-=0, 


ai  —  y 

h. 

C\ 

a^ 

h  -  y 

Ci 

«3 

h 

c^  —  y 

which,  on  being  expanded,  gives  a  cubic  in  y. 


12.    To  find  the  condition  that 

U  =  ax^  +  hx^  +  c^^  +  dx^  +  ^^  +/ 
and  V=ax^  +  I3x'+  yx  +8 

may  have  a  common  factor,  and  to  find  that  factor,  apply 
the  method  of  elimination  exhibited  in  Example  4,  page 
350.     The  result  is  : 

If  ahcdefOO^O, 

OahcdefO 
00  a  bcdef 
a^ySOOOO 
Oa/Sy8000 
OOa^ySOO 
OOOa^yS  0 
0000al3yS 

U'and  Fwill  have  a  common  factor  w^hich,  to  a  constant 
multiplier,  will  be 


358 


DETERMINANTS 

ax  +  b 

a 
ax  +  /3 

a 

0 
0 

c     d     e     f     0 
b     c      d     e    f 
y     S      0     0     0 
/?     y     8     0     0 
a      /?      y      8      0 
0      a      y^     y      8 

If  this  determinant  vanish  identically,  i.e.,  if  the  constant 
multiplier  be  zero,  ^and  "Fwill  have  a  common  quadratic 
factor  which,  except  as  to  a  constant  multiplier,  will  be 


ax'  -\-bx-\-  c 

d 

e     f 

aa;'+;8a;+r 

8 

0    0 

ax  +  li 

y 

8     0 

a 

/3 

7    8 

If  this  determinant  vanish  identically,  ^and  F  will  have 
a  common  cubic  factor  which  will  necessarily  be  "For  V 
divided  by  a  constant. 


Example. 

Let  it  be  required  to  find  the  common  quadratic  fac- 
tor of 

6^^  —  x^--x^+  lOx^  +  l^x  -  40 

and  2x^  +  x^  —  x+l0. 

Following  the  above-described  method,  it  is  found  to  be 


^x''~x-l 

2x'  +  x-  1 

2 


10 

10 

-1 

1 


14 

-40 

0 

0 

10 

0 

-1 

10 

:10 


=  100 


6x'  —  x+7 
2a;'  +  a;-l 
2^-f  1     - 

2x'+    x-1 


14 

10  1 

10      0 

-1     10 

15 

10 

DETERMINANTS. 


]59 


-1500 


2x''~-x+2 
2a;'  +  a;-l 


lb00{2x^-^x  +  D), 


Rejecting  from  this  the  constant  multiplier,  1500,  the 
common  factor  is  2a;^  — 3.r  +  5,  as  may  be  proved  either 
by  actual  division  or  by  evaluation  of  the  determinant  for 
a  linear  factor. 

Ex.  78. 

Apply  determinants  to  solve  the  following  equations  : 


1.    ?>x+1y  =  d>, 
42;  +  9?/  =  ll. 

2.    2x  +  5y=^20, 
Sx-4:y=---7. 

3.    3:r-5y  +  42  =  5, 

1x+2y~?>z=-2, 
4:x  +  ^y—    z=^7. 

4.      X      —y+     z  =  6, 
7x-%y  +  llz=--64:, 
23x-21y  +  24:z  =  lbi. 

|(^  +  2y)=i(3y  +  4z)  =  i(6z  +  5a;), 
x  +  y-z  =126. 

■  6.    l+^u  +  ^z  +  y  +  ^z  =  0, 
\  +  lu  +  ^x  +  iy  +  ^z  =  0, 

24 15 

2«  +  3y     Sx  +  iz 
30        ,        37 


7. 


3z  +  4z 
222 


+ 


Sy  +  Qz 
8 


=  2, 
=  3, 


by  +  9z      2a;  +  3y 
8.    115(13  -  x)  +  719  (y  - 19)  -  590(37  -  z)  ■■ 


■27, 


5(13-a--)+2_37-z 


y-19 


y-21 


=  4. 


360 


DETEEMINANTS. 


9.    u-\-b  —  a(x-\-y),  10,  (a -{-b -j- c)x  =  ay +bz -^c, 

X  -j- b  =  {a -\- 1)  (y -\-  z),  (a+d-\-e)y  —  ax -\-  dz  -j-  e, 

y  +  b==(a  +  2)(z  +  u),  {b  +  d+f)z  =  bx +  dy+f, 

z  +  5  =  (a  +  3)  (^^  +  ^)-  Generalize. 


11.    Given  a:i  =  %i,   y2  =  «i^'i  +  3/i,   ^2  =  %2  +  %, 

2/3==«2^2  +  y2,      ^3  =  %3  +  ^2,      2/4  =  ^3^3  +  3/3, 

prove  that 
^4  =  —  ^1 


-^4 

-1 

0 

0 

0 

0 

0 

1 

—  Os 

-1 

0 

0 

0 

0 

0 

1 

-h 

-1 

0 

0 

0 

0 

0 

1 

—  ftz 

-1 

0 

0 

0 

0 

0 

1 

-h 

-1 

0 

0 

0 

0 

0 

1 

—  «! 

-1 

0 

0 

0 

0 

0 

1 

-h. 

12.    Given  x=-bi/{ai+y),  y=b^/{a^,  +  z),  z  =  h^/{a^+u), 
and  u^bi^/a^]  prove  that 


x  =  bi 


a,  -I    0 

-^ 

bz    ^3    -1 

0      b^     a' 

«!  -1      0  0 

^2  CL2  —  1  0 

0  ^3      as  —  1 

0  0      b^  a^ 


(Take  for  variables  x,  xy,  xyz,  xyzu,  and  eliminate  the 
last  three.) 


Solve 

13.    ax-\-by  —  cz  =  2ab,  14.  {c  +  a)x—{c—a)y~2bc, 

by  +  cz  —  ax=2bc,  (a  +  b)y  —  (a  —  b)z=2ac, 

cz  -{-ax—by=2ac,  (b  + c)z  —  (b  —  c)x  —  2ab. 


DETERMINANTS.  361 


15.  {z-\-x)a  —  {z  — x)b  =  2yz, 
{x-\-  y)h—  (x-  y)c  =  2xz, 
{y+z)c-{y-  z)a-=2xy. 

16.  X  +  ay  +  a^z  -\-  a^u  +  a*  =  0, 
x  +  hy  +  hh  +  h^u  +  ^'  =  0, 
X  -{-  cy  +  c^z  -{-  &u  +  c*  =:  0, 
x-\-dy-\-  dh+  d^u+  d'=  0. 

17.  x-{- ay  +  o?z  + a^u=^  d, 
^-\-^y  +  ^^2;  +  h^u  —  a, 
X  +  cy  +  c^z  -{-  chi  —  5, 
x-\-  dy+  dh+  d^u=  c  ; 

and  if  a,  b,  c,  d,  are  the  roots  of  the  quartic 

V*  —piv^  +P2V'  —p;V  +p^  =  0, 

determine  x,  y,  z,  u,  in  terms  of  pi,  p2,  ps,  p^. 

18.  u-\-    ^+    y+    z^=k, 
au-\-  bx-\-  cy  -{-  dz  —  I, 

a^u  +  b^x  +  c^y  +  c?^2;  =  m, 
a^t^  +  ^^^  +  ^V  +  ^^^  "^  '^• 

19.  Show  that  either  of  the  following  systems  of  equations 

can  be  reduced  to  the  other  : 
(1)     Xi+     X2+    Xs  =  Ui,  (2)  3/1  +  a?/2  +  a'^y,  =  v^, 

axi  +  bx2  +  cxs  =  U2,  3/1  +  by^  +  ^'3/3  =  1^2, 

a^x^  +  ^2:^2  +  c'rr3  =  ih  ]  yi  +  ^3/2  +  ^^3  ==  -^s- 

Generalize. 

20.  There  is  a  certain  rational  integral  expression  whose 

value  depends  on  that  of  x,  and  into  which  x  enters 
in  no  degree  higher  than  the  third.  Its  value  is  4 
when  x=0,  is  9  when  x  =  l,  is  20  when  x—2, 
and  is  49  when  x  —  S.     Find  the  expression. 


362 


DETERMINANTS. 


Solve 
21. 


y^ 


a  +  Tc      h  -\-  k      c  +  k      d  -\-  k 


=  1, 


+  : 


y     I     ^ 


+ 


a-\-n      b -\- n      c-\-n      d  ~j- n 


:1, 

1, 
1. 


22. 


(g  —  ^>)  (g  —  g)  _  Q^ 


23. 


24. 


25. 


bz  +  cy 

c^  +  as;  c  —  a 

c        ^  (c-a)(b-c)_Q^ 
ay-\-bx  a  —  b 

\  X  X  X    =0. 

X  1  c  b 

X  c  1  a 

X  b  a  1 

0  1  1  1 

1  (a^  +  bj         a'  b' 
1         x'        (x'  +  bj         b* 

1         x'  a'        (x'  +  ay 


-0. 


a—bx         X         X 

X         b  —  €         X  X 

X  X  c  —  d     X 

X  X  X  d  —  a 


=  0. 


26. 


a?  b'  c' 

(a  +  xf       (b  +  xf       (c+xf 
(2a  +  xy     (2b  +  xy     (2c  +  xf 


=  0. 


DETERMINANTS.  363 


27.  Determine  a,  5,  and  c  so  that   the   two  systems  of 

equations 

ax  +  by  —  cz  —  /,  a^x  +  ^{y  +  -^^z  =  l^, 

ax  —  by  +  cz  =  m,  a^x  +  ySay  +  y^z  —  mj, 

—  ax  +  by  +  cz=^n;  a-^x  +  fty  +  732:  =  Ui ; 

may  be  satisfied  by  the  same  values  of  Xj  y,  z. 

Apply  to  the  case 

ax-^by  —  cz—    4,  2x—    7/+ 3  2;  =  9, 

ax~by-\-cz—    8,  3:r  +  2y  —  2^  ==^  1, 

—  ao;  +  5y  +  (72;  =  16,  —  r?;  +    y  +    2;  =^  4. 

28.  Solve    2a;  +  3.y-42;_3rg  +  4y-2^_4^  +  2y-3^ 

:r  +  5  5a;  4a:— 1 

_^x^y-z 
6 

29.  Eliminate  x,  y,  and  2;  from 

{a^x  +  b.,y  +  ^>22;  +  c^)/u  =  {b^x  +  agy  +  b^z  +  ^2)/'^ 

=  1  —  CiX  —  c^y  —  c^z  ; 

ux -\- '^y  -\- '^'z  ^=^  1. 

30.  Determine  a,  given 

^  +  y  +  2;  +  -u;  —  0,  ax  +  by  +  cz-{-dw  =  0, 


a^b^c^e        '  a'^ 


c' 


31 .  Solve  li  (liX  +  m^y  +  Wi2;)  =^  am/  +  bui^, 
4  (4^  +  "^hy  +  ^22;)  ^  am2^  +  J7^2^ 
4  (4a:  +  m-sy  +  7232;)  ==  ama^  +  bn^^ 

44  +  ^zmg  +  712^3  —   44  +  ^3^1  +  ^3^1 

=  44  +  ^iTTZa  +  71x^2  =^  0. 


364  DETEEMINANTS. 


32.  If  Xi,  3/1 ;  X2,  ^2 ;  ^3 J  3/3  are  the  values  of  x  and  y  that 

satisfy  each  possible  pair  of  the  equations 
aiX  +  biy  +  Ci  =  0, 
«2^  +  hy  +  ^2  =  0, 

«3^  +  %  +  ^3  =  0, 

prove  that 

I  1  ^'23/3  I  =  1  <^A^3  P-^   l\  CcA  I   I  <^253   I  I  CtA  \]' 

33.  The  equations 

x  +  S7/  +  5z  +  Su  =  S4:,  x+27/  +  5z  +  4:u  =  S6, 
x+  y  +  2z+  u  =  lS,  x  +  3y  +  8z  +  5u  =  51, 
have  for  sole  solution  x  =  l,  y-—2,  z  =  S,  t^  —  4,  but 
.  on  attempting  to  find  the  value  of  u  by  indetermi- 
nate multipliers,  on  adding  together  the  equations 
multiplied  respectively  by  1,  a,  13,  y,  and  equating 
to  zero  the  coefficients  of  x^  y,  and  z  in  the  resulting 
equation,  we  obtain  the  incompatible  equations, 

1+    a+    13+    y-O, 

3+     a  +  2i8  +  3y-0, 

5  +  2a  +  5i8  +  8y-0. 
Explain  the  paradox. 

34.  Eliminate  x,  y,  and  z  from 

ax -{- by  -{-  cz  —  1  =  hiX -{•  a^y  —  z  +  c 

=  CiX  —  y-\-aiZ  +  h=-~x-{-ciy-{-biZ-\-a  =  0, 

35.  Eliminate  u,  v,  w,  x,  y,  z  from 

aiU  +  biV  +  CiW  =  0,  ttir?;  +  /3iy  +  yiZ  =  u, 
a^u  +  b<iV  +  c^w  —  0,  a^x  +  ^iy  +  y22;  =  v^ 
a^u  +  h^v  +  (^s-w;  ^-  0  ;        as^  +  ^iy  +  yg^  =  -m;  ; 

and  prove  thereby  that 

I  ai&,^3  I  X  I  ttjftys  I 

=  I  aiai+^>ia,+  Cia3,  afi^+b.^.^+cSz.  a.,y^+hy>,+  C-,y^  |. 


DETERMINANTS. 


365 


36.    Eliminate  u,  v,  and  w  from 

au-\~  hv  -{-  gw  =  \u^ 
hu  +  hv  +fw  ~  \v, 
gu  +fv  +  ciu  —  Xw  ; 

and  w,  V,  w,  x,  y,  z  from  the  three  preceding  equa- 
tions combined  with  the  three  following : 

ax  -{-  hy  -{-  gz  —  u  ~  \x, 

hx+by  +fz  =v  ~Xy, 

9^  +fy  +cz^w  —  \z] 
and  reduce  the  two  resultants  to  the  same /orm. 


37.  Eliminate  first,  x,  y,  z,  second,/,  g,  h,  from 

aw  -\-  hy  —  gz  =0, 
hw  -\-fz  —  hx  =  Oj 
cw  -\-  gx  —fy  =  0. 

38.  Show  that 

<^l<^2  +  ^1^2  +  <?l<^2  <^-l    +  b.^  +  C^ 

=  1  <^A  P  +  I  ^1<?2  P  +  ^1^2  f. 

39.  Prove  that 

I  «i^2^3  r + 1  ^i^2<^3  r + 1  ^i<^2^3  r + 1  ^i^^^s  r 

a2ai  +  &2^1  +  C2^'l  +  ^2^1>  «2^  +  V  +  <^2^  +  ^2^'  <^2%  +  ^2^3  +  ^2^3  +  ^2'^3 


agai  +  ^gJ^  +  CgCi  +  cZgC?!,  a3a2  +  &3&2  +  ^3^2  +  <^3^2» 

Generalize. 


agHV+CgHc^a^ 


40.    Let  Ai  =  |ai^2C3l,  A2=|ai^2y3|,  and  AjAa^  l^i^aC'sl, 
then  will    0      ao     ^o     7o 

<^0  ^1  -^2  -^3 
&0  -^1  -^2  -^3 
(?0  v^l         ^2        ^3 


366 


DETERMINANTS. 


State  this  tiieorein  in  tlie  cases 

r.     Ai  =  A2,   ao--ao,   Po=--ho,  yo=Co] 

2°.     A,  =  A2,   ao  =  ao,   l3o=--bo  =  yo  =  Co=^0. 

41 .  Given  Ui  =  UiX  +  biy  +  CiZ  -f  di  =  0, 

U2  E=  a-^x  +  b^y  +  C22;  +  r:?2  =  0, 

^^3  =  a3:^;  +  %  +  (?32;  +  (^3  =  0,  1^  (1) 

U4,  =  a^:?;  +  b^y  +  C42;  +  c?4  =  0, 

2^5  =  a^x  +  553/  +  c^z  +  J5  ---  0. 

1°.  Determine  tlie  value  of  x  that  will  satisfy 

ttlt^l  +  a2W2  +  <^3^3  +  Cl4^i  +  <^5'^5  "^  ^)   ^ 
CiUi  +   ^2^2  +   ^3'2^3  +   <?4^4  +   ^5^5  =  0.    J 

2°.  Eliminate  z  from  the  group  of  equations  (1),  taken 
two  by  two  in  every  possible  way  ;  from  the  result- 
ing ten  equations  form  two  equations  by  a  method 
similar  to  that  by  which  the  set  (2)  was  formed  from 
the  group  (1) ;  and  determine  the  value  of  x  that 
will  satisfy  these  two  equations.  Show  that  this 
value  of  X  is  the  same  as  the  value  obtained  by  the 
solution  of  the  set  (2). 

Apply  the  preceding  to  the  equations 

X-    y  +  2z-   3  =  0, 

3:r  +  2y-5^-    5  =  0, 

4a;+    2/  +  42;-21  =  0, 

a:  +  3y  +  32;-14  =  0. 

Generalize 

42.  Eliminate  x,  y,  z,  and  It  from 

aiO:  +  /?,y  +  y^z  =  0,     a^x  +  ^^y  +  JiZ  =  0. 
a^x  +  b-^/  +  h^z  ^  bsX  +  a^y  +  ^i^  ^  h^  +  ^ly  +  cisZ 
tti  +  02^  ^1  +  ^2^  yi  +  7-2^ 


DETERMINANTS. 


367 


43.  Eliminate  x,  y,  z,  it,  ki,  and  ^2  from 

ai^  +  /^ly  +  Ti^  +  Sit6  =  0, 
a^x  4-  fty  +  y,z  +  8^2^  ==  0, 

"3^  +  Ay  +  732;  +  ^3^^  -=  0, 

ax  +  Ay  -\-  gz+  lu  __hx  +  by  +fz  +  '^^^ 

gx  -\-fy  -\-  cz-j-nu  _  Ix  +  my  -\-nz  +  du 

yi  +  y-ih  +  y^h  ^i  +  ^2^1  +  ^3^2 

44.  If  Q'l^  +  ^ly  +  ^1^  ^  0^2^  +  ^23/  +  C2Z 

U  V 


w 


then 


AiU  +  A2V  +  A3W      BiU  +  ^2'y  +  ■^3'^^ 


in  which  ^1,  ^2,  etc.,  are  the  inverse  elements  of  ai, 
(^2,  etc.,  with  respect  to  |  aJ)2Cz  |. 

45.  Eliminate  x,  y,  and  z  from 

^  +y'  +2'  ==^, 

(:?;-a)^+y^  +2^^  =(k-d)\ 

(x~a,y  +  {y-b,y  +  z'  ^{k-d,)\ 

(X  -  a,y  +  (y  -  ^2)^  +  (^  -  ^2)^  =  i^-  d,)\ 

46.  Apply  the  results  of  Examples  1  and  2,  pages  348  and 

349,  to  prove  that  if  the  value  of  a  determinant  be 
zero,  the  determinant  may  be  transformed  into  an- 
other of  the  same  order  in  which  all  the  elements  of 
a  row  or  of  a  column  shall  be  zero. 


Transform 


4      3 


1 


2  -3 

3  4 

6       5-2-5 

10  -12      3       4 

into  a  determinant  of  the  fourth  order  with  a  col- 
umn of  zeros. 


368 


DETERMINANTS. 


Similarly  transform 


47. 


^  2  -4 
2-13      5 

B  7-11 
1    -9       3 


+  26;r+2/3/+(^:-0, 
-0, 


Apply  the  above  to  prove  the  rule  for  the  multiplica- 
tion of  determinants. 

Eliminate  a,  h,  c,  d,  ^,/from 

ax  +    lipcp  +3/)  +  cy^  +    e  ■\-  f]p 

a    +    h{2jp-^^q)-\-c{f-\-yq)  +  f(i 

h{^q-\-xr)  ■\-  ci^jpq^yr^  +  fr 

5(4r  +  :i;s) +c(4pr  +  3^'+y5)  +  /s 
Z>(5s+:r^)  +  c(5;9S  +  10^r+yi5)+  jt 

and  evaluate  the  resulting  determinant. 
[Omit  the  first  three  equations, — this  eliminates  a,  e, 

and  d\  in  the  remaining  three,  take  for  variables 

5  +  cp,  cq,  and  Z):?:  +  cy  +/•] 

48.    If-^    +^2;  +  (7y  +  i)z^  =  0, 
Az-B   +  &  +  i)i;  =-  0, 

Jw+^v  +  C^^--^    ==0, 
then  will 

^2  ^2 


-  w^ -\-2iXvw      l  —  u^—y^—w^-^-^uyiv 

Q2  JJ 


l-~u'^  —  v'^—z^-\-2uvz      1- 

49.    Eliminate  u,  v^  and  w  from 
w^i  +  vyi  +  'w;2;i  =  0, 

0. 


-y'^  —  z^  +  2  xyz 


a 

h 

9 

u 

h 

b 

f 

V 

9 

f 

c 

w 

u 

V 

w 

0 

DETERMINANTS. 


369 


50.    Show  that  the  system  of  equations 

a(b  —  c)  .  b  (c  —  a)  .  c(a  —  h)  __r. 
a  —  a  0  —  p  c  —  y 

a  —  a  13  — b  y  —  c 

is  satisfied  by  either 

a  —  b-{-l3  —  a  —  b  —  c  +  y  —  l3  —  c  —  a-\-a  —  y, 
or      a^y  =  aby  =  a/Sc. 

Tf    ^^  +  cty  —  cz_cx  —  b7/  +  az  __  —  ax  +  cy  +  bz 
d'  +  h'      ~       d'  +  d'      ~  b''  +  c^ 


then  will 


X 

y 

2, 

c 

a 

b 

b 

c 

a 

0. 


52.    Eliminate  x,  y,  and  z  from 

a(.^-l)  +  5(y-l)  +  c(.-l)^0, 
x  +  y  +  z  =  l, 

-0. 


X 

y 

z 

c 

a 

b 

b 

c 

a 

[Reduce  the  determinant  to  the  form 

ax  -\-by  +  cz      x  +  y  -\-  z     =0.] 
ab  -\-  be  -\-  ca     a-\-b  -\-  c 


53.    If  Ix -{- TYiy -\- nz    =0, 

ax  —  by   —  cz    =  u^ 

—  ax-\-by   —  cz    =v, 

—  ax  —  by-\-cz    =  w, 
aux  +  bvy  +  cwz  =  1, 

then  will 

m'  +  n'    _    n'  +  P 


I    771    n  \  =  0, 

X    y     z    \ 


P  +  m' 


(bn  +  cmy      (cl  -\-  any      (am  +  biy 
==  x"^  -\-  ?/  +  2;^ 
and  4  d'b'cXx'+y'+zy-(a'+b^+c'')(x'+y'+z')+l=  0. 


370 


DETERMINANTS. 


64.    Exhibit  in  a  single  equation  the  result  of  eliminating 
u,  X,  y,  z  from 

ax  -\-  hy  -{-  gz  =  aiii  +  XiX, 

hx  +  by  +fz   =  b,u  +  Ajy, 

9^  +fy  +  cz  =  Ciu  +  Aiz, 

ai^+  h^y-\-  (?i2;  —  0  ; 
and  V,  Xi,  3/1,  Zi  from 

axi  +  hyi  +  gzi  =  aiV  +  Xj^, 

hxi  +  Z>yi  +  /2;i  =  b,v  +  Ajy, 

^^1  +  fyi  +  G^\   =  OiV  +  X2Z, 

ctiXi  +  b,yi  +  (?i2;i  --  0. 

55.    If  ai  +  biy  +  CiU  =  a2V  +  b2+C2Z  =  asX+biW  +  Ci—s, 

and  wo;  —  vy  =  wz  =  l^ 
then  will 


«!  —  s        5i 

a. 


^3      <?3  —  s 


66.    Eliminate  A  from 


JL 


+  - 


■A.      6  —  A.      c  —  A 


-  + 


yy 


a— X     6— X 
57.    Eliminate  A  from 


+  - 


=  0. 

=  0, 
=  0. 


■  + 


x 


a  — X     b  —  \ 


■  + 


yy 


X    ^-x 

,        ZZ^       ,       1^16' 

i -r-T  • 


=  0, 


a— A ' b—\     c—\     g~\ 

58.    Eliminate  u,  v,  and  t^  from 

p^/  u  +  5' V  V  +  r^  /  w  =  0, 
P  /  U  +  m^/  V  +  ny  tc'  =  0, 
P (^  -  r)  z^  +  Q  (r  -;?)  i;  +  i?  (;?  -  ^)  t^  r:=  0. 


DETERMINANTS. 


371 


■     59.    If 


60. 


61. 


oi,-a,y  +  {p,-p,y  =  Ti  +  n\ 
oi,-a,y  +  {p,~iB,y  =  Ti  +  n\ 


then  will  Vi  ^  +  ra"^  +  7-3"^  +  r^ 


=  0, 


:0. 


and  ^i^r'  +  A^rf''  +  A^r 

in  which  A  is  either  a  or  ^. 

Eliminate  u,  x,  y,  z  from 

ai6  +  6:?;  +  ^y  +  <^2;  =  0, 

a  /u-\-h  /  x-\-c  /y  -\~  d  j  z 

Eliminate  u,  v,  w,  x,  y,  z  from 
Ix  +  my  -\-nz  =0, 
lu  +  m-y  +  9216'  =  0, 
fyz+gxz  +  hxy  =  0, 
ux     vy     wz 

62.  Eliminate  x,  y,  z  from 

ax  +  hy  +  cz==/{ax)  +  /{hij)  +  /{cz), 

^{x/u) + v(yA) + V(  V^)  =-  0^ 

^/(l  -  he)  +  y/(l  -  CO)  +  V(l  -  «^)  =  0, 
and  reduce  the  resultant  to  the  form 
Au  +  Bv+Cw  =  0. 

63.  Eliminate  x  and  y  from 

(1  +  x)/(ia  -y)-=  /a  +  x/a, 
{l  +  x)/{h-y)=--/h  +  x/p, 
(1  +  x)/{c  —y)  =  /c  +  x/y. 


372 


detehminants. 


64.    Eliminate  x,  y,  z,  s  from 

(s  -  y)  (s  -  2)  -=  ayz, 
(s  —  z)(s  —x)  —  bzx, 
{s  —  x){s—y)=-cxy, 
x+,y  -{-  z'=2s. 

xyz  —  a-]-  yz(y  -\-  z)  =  b  +  zx(z-\-  x) 
=  c  +  xy(x  +  y), 
show  that  4  (xyzY  —  (ah  -{-bc  +  ca)  (xyz)  +  abc  =  0. 

Determine  X,  fi,  v,  given 

a  —  a    b  —  /3    c  —  y 


65.    Given 


66. 


I          m          n 

IX  +  TUfji  -\-nv^=0, 
V^  +  /x^  +  v^  =  l. 

67. 

Find  u,  V, 

w,  given 

lu  +  mv  +  nw  =  0, 
liU  +  yiiiv  +  TiiW  =  0, 
ii''  +  v'  +  w''  =  1. 

:0. 


68.      If 


show  that 


A  (ax  +  hy+gz  +  Z^^)  +  fi  (hx-{-by'\-fz-\-7}iw) 

+  v(gx  +fy  +  CZ  +  nw)  ^0, 
aX -^  hjji -{-  gv  =  uX, 
hX  +  Z>/x  +/i/  =  w/x,, 
^^  +/)".  +  (?v  -=  uv, 
iix-[-lw      ,     uy-\-r}iw    _^    uz  +  nw    _ 


•  +  - 


■  +  ; 


0. 


69. 


(a-u)f-gh      (b-u)g-fh      (c-u)h~Jg 

Eesolve  a  system  of  three  equations  in  three  unknowns 
of  which  one  equation  is  quadratic  and  two  are 
linear. 

70.    Apply  the  method  of  Example  9,  p.  355,  to  resolve 

x'-\-f^z'--^\ 
UiX  +  hiy  +  CiZ  =  UX, 


DETERMINANTS.  373 


a^x  +  h^y  +  c>2Z  =  uy, 
«3^  +  hy  +  <?3^  ==  uz  ; 

and  compare  the  values  of  u  given  by  the  resolution 
and  that  obtained  by  eliminating  x,  y,  and  z  from 
the  last  three  equations. 
Generalize. 

71.  Eliminate  u^,  Vi,  u^^  V2  from 

X1U2  —  x^Ui,         ax  I  +  du2  —  hx2  +  cui, 
yiV2  ^  y-iVi,  av2  +  dyi  =  bv,  +  ct/^, 

(ad  —  be)  X1U2  =  X2  yt  —  x{yx  +  iiiVx  —  u^v^, 

72.  Eliminate  x,  y,  and  z  from 

x-\-y-\-z^^, 

ax"  +  by''  +  cz^  +  2fyz  +  'I.gxz  +  2  hxy  =  0, 
a^x^  +  ^2^'^  +  C'i^^  +  3  a2^V  +  3  '^a^''^^;  +  3  b^xy'^ 
+  3  Z>3y^2;  +  3  c^xz!-  -f  3  (^z^^^^  +  6  (frrya;  ^  0. 

73.  Eliminate  x,  y,  and  z  from 

^2  _|_  ^2  _  2^:r?/  =  0, 

and  assuming  the  resulting  relation  to  hold  among 
g,  A,  and  Ic,  find  the  H.C.F.  of  the  functions 


m'  + 1;'  - 

-  'iguv 

-(1- 

-/). 

?;^  +  w''  - 

-  2  At/'ty 

-(1- 

-A'O, 

wHw'- 

-  2  fooM 

-(1- 

-^'0. 

74.    Eliminate  x,  ?/,  and  2;  from 

(/i  —  '^0  y^;  +  gxzx  +  /^i-ry  =-  0, 
f^y^  +  (^2  --  '^)  ^^  +  ^^^2^^  =  0, 
f^yz-\-g^zx-\-{h^-u)xy  =0. 


374 


DETERMINANTS. 


75.    Eliminate  x,  y,  and  z  from 

a,o^  +  biy''+c^z^  +  2{f^-u)yz-\-2g^zx+2h^xy  =  0, 
a^jXp-  +  b^y"^ + c<]2^  +  2/23/2;  -}-2(g2  —  i^)zx-}-2  h^xy  =  0, 
a3x''  +  b,y^+c.,z''  +  2fsyz  +  2y^zx  +  2(hs-u)xy  =  0. 


76.    Expand 

x'  ~  a'       7/           z' 
x'       f  -  b'       z' 
x^           y^       z^  —  c^ 

-0. 

77.    Expand 

a'~f-z'           y-                   z' 
x"           b''-x^-z^           z' 
x^                   y^           &  —  x^  —  y^ 

=  0, 

and  reduce  the  expanded  equation  to  the  form 

av        z>y    ,     c\^ 

=  0 

a^-s^      h'-s'      c'-s' 
s'^  =  x"^  +  y^  -\-  z\ 

78.    If  a,  (3,  y  be  the  values  of  u  satisfying 

x-{-  a  —  u         y  z 

X  y  -{-  b  —  u  z 

X  y  z  -]-  c  —  u 

and  if 


th( 

d  —  y  —  z         y                z 
X         b  —  z  —  x          z 
X                 y          c  —  x 

m  will 

(cT-a)(cT~l3)(cT-y)  =  0, 
in  which 

< 

T  =  a  +  (3  +  y-~a  —  b  —  c. 

=  0, 


0, 


79.   Solve  of'  +  2x'y  +  2a;y  (y  -  2)  +  y'  -  4  =  0, 
x'  +  2xy+2f~5y  +  2=-.  0. 


DETERMINANTS. 


375 


'      80.    Solve 


81.    Solve 


x^  +  Sxy^  —  6x7/  ~x-{-y^  ^-=0, 

3 :?; V  -  3  ^'  +  y'  -  3  y^  -  3/  4-  3  -  0. 


-6. 


1)^ 


(x  +  l)(y+l) 
[Transform  by  u  =  x-{-  7/,  v=  xy,  and  eliminate  u.'] 


82.    Solve 


^^  +  a;y +  /  =  17, 
x  +  xy  +  y^b, 
[Transform  by  u=^x-\-y  -\-  xy,  v  =^  {x-{-  y) xy']. 

83.  Solve  x'  +  y''  +  z'  =  (a  +  u)\ 

(a-xy  +  f  +  z'  =  (/3  +  u)\ 

(a,  -  xf  +  (h,  ~  yy  +  z'  -  (y  +  u)\ 

(a,  -  xj  +  {h,  -  7/)^  +  (6',  -  zj  -  (8  +  u)\ 

84.  If    a  =  ^iX  +  a^/^  +  <^3^j     ^  =  ^A  +  ^2)^  +  ^si', 

ax^  +  )8a;i'  +  y:ri  +  8  =  0, 

a:r3^  +  )Sa;3^  +  ya;3  +  8--0, 
in  which  x^,  x^,  and  ^^a  are  the  roots  of 

x^  -\-px^  +  qx  -\-r^=^0, 
show  that     1      p      g     r     =0, 

d, 
d. 

What  does  this  equation  become  if  o^i  =  ^^2  ? 
What  does  it  become  if  x^  =  x^  =  x^? 


1 

? 

9 

«! 

^1 

Cl 

a. 

b. 

C2 

(h 

6a 

Cs 

85.    If 


aiXi^  +  h^i^  +  <^i^i  +  di_  a^x^  +  h^x^  +  c^x^  +  6^2 

aa^l^Z^  +  h^'i     +  ^1^2  +  <^2  <^2^2^  +  ^2''^2'^  +  <?2^2  +  <^2 

a-iXi  +  ^3^2'  +  <?3^2  +  <^3 


376 


DETERMINANTS. 


show  that  Xi  and  X2  are  the  roots  of  the  quadratic 


x' 

—  X 

1 

A 

B 

c 

B 

C 

D 

-0, 


in  which  A,  B,  C,  and  D  are  the  four  determinants 
of  the  third  order  that  can  be  made  from  the  array 

'«!  hi  Ci  di 
a2  h^  C2  c?2 

(^3     0>3     C^    d^. 

86.    If  u,^+  vi'  =  u^  ■\-v^--=-\,  then  both 

{ a'  {x  -  u^^  +  h\y  —  v^f  ]  (xu^  +  yv^  —  1)' 
-=-\a^{x—u^''-\^'b\y-v:^''\{x%ii-^yvi-Vf 
and  \  a^(x  —  Uif  +  h^(y  —  ViY  \  { (xv2 — yu^'^  —  {x  —  Wa)^ 

-={a\x— u^y + bXy - v^f  |  { (xvi  —  yu^f 
~{x-u,y-(y  —  Viy\ 
are  satisfied  by 


X      y      I 

Ui      Vi      1 

U2    V2     1 


=  0. 


87.    If  rAh  (h,'  +  P)=  r, A3A1  (A^^  +  P)  =  r.hA (V  +  P). 
then  will 

-0. 


Ai 

A2A3 

^2^3 

h. 

hA 

r-,i\ 

h 

hji2 

i\r2 

88.    If  ?'  +  ^'  =  ^  +  ^^ 


xxi  ■  y.Vi  ■    g^^ 
a'^  "^  />^  "^28,^ 


^2    I  .V2  1 

a^  "^  6'^         ' 


2  81^ 


\X2  ,  y,y.,  ,    ct   _  1 
^2  ^  ^2  ^28' 


a' 


DETERMINANTS.  377 


show  that  j  1     X      y      _  ahaaia^ 
1     -^1     Vi    ~  ~2MA 

89.  Given  that  aia;'  + 2 Z>ia.-  +  ^i--0  and  a^,x'  +  2h,x  +  c.,-=-0 

have  a  common  root,  determine  it. 

Apply  to  case  of 

I  990^' -441 5; -5390  =  0. 
1  825  x^  -  428  X  -  4620  =  0. 

90.  Given  that  ax^ -\-?>bx'^  -\-?>cx-{-  d  has  a  square  factor, 

find  it. 

Apply  to        2940  x^  +  812  x""  -  8385  x  -  6300. 

91.  Determine  the  condition  that  ax^-\~?>bx'^-\-?>cx-{-  d=^0 

and  ax^ -\-2px  +  y  =  0  shall  have  a  common  root, 
and  find  it. 

Apply  to         I    30^'  +  :^' +  35^ +  204-0, 
1 110^' -23a; -357  =  0. 

92.  Determine  the  condition  that 

ax^  +  4  hx^  -{-Qcx^  +  ^dx-{-  e 
and  aa;'  +  3^.r'  +  3y^  +  S 

shall  have  a  common  linear  factor,   and  find   the 
common  factor. 

93.  Determine  the  condition  that 

ax^  +  4:bx^ -\- ^ cx'^  +  4l dx  +  e 
and  ax'  +  ^Px^+Q>yx'+4:^x  +  € 

may  have  a  common  quadratic  factor,  and  find  the 
common  factor. 

Apply  to  r  60 .r*  -  4ru^  +  37ri;^  -      :r  +  28, 

I  ^Qx'  +  ?>x^  +  Mx"  +  22^;  +  56. 


378 


DETERMINANTS. 


94.  Determine  the  conditions  that 

aiX^  +  2  biX  ~{-  Ci  =  0, 
a2x'^-\-  2K^x  +  c,i  =  0, 
a-iX'^  +  2  b^x  +  c^  —  0, 
shall  have  a  common  root. 

95.  -Determine  the  condition  that 

ax'  +  Ux'  +  Qcx'  +  4:dx  +  e  =  0 
shall  have  two  equal  roots. 

96.  Determine  the  conditions  that 

ax'  +  Ux'  +  6cx'  +  4:dx  +  e  =  0 
may  have  three  equal  roots. 

97.  Determine  the  conditions  that 

ax^  +  5hx'+10cx'  +  10dx'' +  bex  +  g=:0 
may  have  three  equal  roots. 

98.  Determine  the  conditions  that  ax^hy  may  be  a  com- 

mon factor  of 

a^x^  +  2  h^xy  +  c{y'^ 
and  a^x^  +  3  h^jX^y -f  3  c<iX.y'^  +  d^^if. 

99.  Determine  the  remainders  in  the  process  for  finding 

the  H.O.F.  oifixY  and  J^(:l:)"^+^ 

100.    If  ^"»  +  l^x"^-^  +  ^2^"^-^  + be  divided  by 

x""  +  a^x"""^  +  a<iX''~'^  + , 

then  will  the  coefficient  of  the  rth  term  of  the  quo- 
tient be 

{-\y-^    110       0 
h^      a,      1       0 

^2        a2        «!        1 
63         ag         ^2         <^l 


&^_i   a^_i  a^_2  a^- 


DETERMINANTS.  379 


■      101.    For  what  value  of  a  will 

and  ax^  —  5.'r +  4a 
have  a  common  factor? 

102.  For  what  values  of  3/  will 
x^  —  4  x'^y  +  xi/  —  {y  —  Vf 

and(rr-l)'^y  +  (3/-iy 
have  a  common  factor  ? 

103.  Find  the  relations  that  must  hold  among  a,  h,  and  c, 
that 

ax^  -\-hx-{-  c 

and  a(l  —  c):i;^  + ^^(1"^)  +  ^*^?  ^  + ^ 
may  have  a  common  factor. 

104.  If  2x^  —  x^  -\-  a^Q  and  a;"  —  :i:  +  Z)  —  0  have  a  com- 
mon root,  then  must 

(4  ^)  -  1)  (2  ^)^  ~  a)  +  (a  +  2  B'  -by  =  0', 
and  if   ^  +  2  ==  0,    find   the  values  of  a  and  the 
resulting  equations. 

105 .  If  a^x"^ -}~2biX  +  Ci  =  0  and  ^2^^  +  2 ^2^  +  ^2  —  0  have 
a  common  root, 

(aiCi—bi^)x'^+{aiC2+Cia2—2bib2)x+(a2C2~h^)=0 
will  have  equal  roots. 

106.  If  a.x''  +  2bj,x  +  c,  =  0  and  a,x^  +  2b.,x  +  a,  =  0 
have  a  common  root,  their  other  roots  are  given  by 

<^i«2 1  biC^  \x'^  +  \  aiC2  P^  +  C1C2 1  aiS2 1  —  0. 

107.  Eliminate  x^,  x^,  x^  from 
«i  ('^\  +  ^3)  =  —  2  Z)i,     aiO^i^Ta  =  ^1, 

(^2  (0:2  +  0:3)   =  —   2  &2,         <^2^2^3  =^-  Cl- 

Form  the  quadratic  whose  roots  are  x^,  x.^ ;  and  the 
cubic  whose  roots  are  Xi,  X2,  x^. 


380  DETEKMINANTS. 


108.  Find  the  condition  that  must  be  fulfilled  in  order  that 

shall  have  equal  roots  in  u. 

109.  Find  the  condition  that 

x"  -2 


+  -^  +  ^-=1 


u-\-  d^      u^b'^      u-\-  & 
may  have  equal  roots  in  u. 

110.  If  ax^  -\-  Mx^  -\-  ^cx^  -^  ^:dx  -^  e^^   have    a   double 

root,  it  will  be  given  by 

[For  the  values  of  7  and  7,  see  (28)  and  (29),  p.  305.] 

111.  Shovv^  that  if  the  quartic 

ax'  +  4^:r'  +  ^cx^  -\-^dx^e^^ 
have  three  equal  roots,  then  will 

ad-lc   _ae  +  2bd-?>c\__      S(be-cd) 
2{ac-b')  S(ad-bc)         ae  +  2bd-Sc' 

^2(ce-d')^ 
be  —  cd 
and  prove  that  these  equations  are  equivalent  to 
7-7-0. 

112.  Show  that  the  conditions  that  the  quartic  in  111  shall 

have  three  equal  roots  are  the  same  as  the  condi- 
tions that 

ax''  +  2bx  +  c=^0, 
bx''  +2cx  +  d=0, 
ex'  +  2dx+e=0, 

shall  have  a  common  root,  and  express   them   as 
determinants.     (See  problems  94  and  96  above.) 


DETERMINANTS.  381 


113.  Find  the  relation  that  must  exist  between  g  and  h  in 

order  that 

z^  -  10^  V  +  12  h'z  +  5^-^-0 
may  have  a  pair  of  equal  roots. 

For  what  values  of  z  will  each  of  the  following  equa- 
tions have  a  pair  of  equal  roots  ? 

114.  x^  ■\-{z-  \)x'  -\-iz-  8)x  -6(z-2)  =  0. 

115.  x'  +  2zx'  +  (z'  -5z-  75)x  -  5(z'  +  5^  -  50)  -  0. 

116.  x'  +  (3^  -  2)x'  -  (6z  +  16)x-  4:bz  =  0. 

117.  x'  —  2(z  +  2)x'  +  {8z  +  3)x-6z  =  0. 

118.  x'  +  (z  +  (j)x'  +  (4;^  +  11)^;  +  3  (^  +  2)  =  0. 

119.  x'-~^x  +  2z  =  0.       122.  x'+^zx-(l-zy-4:z'  =  0. 

120.  x'-^x'  +  z'  =0.       123.  2a;^-3^;r  +  /  +  l  =  0. 

121.  x''~ozx  +  z'-=0.       124.  x'—9x'  +  ^x  +  z  =  0. 

125.  X'  +  4:X'  +  4:4:X'-9QX  +  Z  =  0. 

126.  \x'-(l-z)x'lb'-^'z(l-zy  =  0. 

127.  x'  +  x^  +  x''  +  z  =  0. 

128.  Find  the  relation  that  must  hold  among  the  coeffi- 

cients that 

x'  +  6bx'  +  156?:?;'  +  20dx'  +  15  ex' +  6fx  +  g' 
may  break  up  into  the  cubic  factors 

x^  +  3  ^1^^  -{-obiX-\-g 
and  x^  -{-3  a^^x'  +  3  h^^x  +  g. 

129.  Show  that  the  discriminant  of 

a  {x^  +  3/^)  +  ^  {yx^  +  v~^  y^)  xy 
+  c  (v^x  +  v~'^  y)  ^y 
is  a  rational  integral  function  of  a,  h,  c,  and  (t'^+'y"^) 
and  of  the  second  degree  in  the  last  of  these. 


382  DETERMINANTS. 


130.  Prove  that 

(A^  -  ah)  {ax'  +  hf  +  2fy  +  2gx  +  2  hxy) 
--aP--hg'  +  2fgh 
can  be  resolved  into  linear  factors,  and  find  them. 

131.  Show  that  if 

a  +  h  +  c:==^0 
and  a^b  +  b'^c  +  c^a  +  ^mabc  =  0, 
then  will 

ax'  -\-  hy'  -\-  cz'  -\-2 {mc  -\-  d)yz-\-2 {ma  +  h)xz 
+  2  (m^  +  c)  xy 
be  the  product  of  linear  factors.     Find  them. 

132.  li  x'  +  2 hxy -\-y'  —  bx—*ly-\-Q  has  linear  factors, 

find  them. 

133.  For  what  values  of  A  will 

2{x'  +  by'  +  z'  -yz-1xz  +  2xy) 
~X{x''  +  ^y''  +  2z') 
be  resolvable  into  linear  factors  ? 
Find  the  factors  in  each  case. 

134.  For  what  values  of  A  will 

80:?;'  +  8y'  —  ^z'  -  ?>yz  +  bzx  +  33rry 
-  A  {x'  +  y'  +  z') 
be  resolvable  into  linear  factors  ? 
Find  the  factors  in  each  case. 

1 35.  If         {l^x  +  m^y  +  niz)'  +  {l^x  +  m^y  +  n^zf 

+  {kx  +  m.,y  +  n.,zf  ^  :r''  +  y'  +  z\ 
show  that 

80.r'  +  8?/  -  4^2  -  32/0  +  bzx  +  33:ry 
=  a  {l^x  +  mi2/  +  7ii2;)'^  +  p  {l^x  +  w^y  +  n.,zy 
+  y  {kx  +  m^y  +  n.iz)\ 
in  which  a,  p,  and  y  are  the  values  of  A  found  in 
the  preceding  problem. 


DETERMINANTS.  383 


136.  Find  the  condition  that 

{x'+yz){h  —  cXl  +  Be)  +  (y'  +  zxXc  -  a)(l+  Jcca) 
+  (z^  +  xyXa  -  ^)(1  +  kab) 

may  break  up  into  linear  factors. 

137.  Find  the  condition  that 

+  (as:?;'  +  h^y''  +  c.,2;'  +  2f^yz  +  2  (/..^z  +  2  h^xy) 
shall  be  resolvable  into  linear  factors. 

138.  Determine  k  so  that 

4:x''  -  9/  —  2^'  -  ^yz  +  2^:2  +  ?>xy 
+  h{x-?>y-\-z){x  +  y-~bz) 

may  be  resolvable  into  a  pair  of  linear  factors. 

139.  Find  the  condition  that 

Ic  {a^x^  +  3  hix'^  +  3  Ci^r  +  d^ 
+  ^(^2^'*  +  3  h,x^  +  ?>c^x  +  d^) 

shall  have  a  square  factor,  and  this  condition  being 
fulfilled,  find  the  square  factor. 

140.  Given 

xyd'  +  yyb'^-zyc'^o, 

Ix  +  my  +  712;  =  0, 
find  the  condition  that  the  ratios  x:y\z  shall  be 
each  single-valued. 

141.  Given 

fV  +  gY  +  hh''  -  2ghyz  -  2fhxz  -  2fgxy  =  0 
and  Ix  +  "iny  +  nz  =  0, 

find  the  condition  that  the  ratios  x\y\z  shall  be 
each  single-valued. 


384 


DETERMINANTS. 


142.  If  ax-\-l^l/  +  yz  =  0, 

and  f/x  +  g/y  +  h/z  =  0, 

and  if  the  ratios  x  \y  \  z  are  each  single-valued, 
then  will 

143.  Find  the  condition  that  if 

ax"  +  hy''  +  cz"  +  2fyz  +  2gxz  +  2hxy-=  0, 


nd 

a       h      g       Q   '    z     ~y            0 
h       h     f     -z     0       X              0 

g      f      c       y     —X     ^              0 
0     -2     y       1       0       0              I 

z       0    -X    0       I       0             m 

-y     X      0       Q       0       I              n 

0       0      0        I      m      n     P  +  m'  +  ri' 

le  ratios  x  \y  \z  shall  be  each  single-valued. 

144.  Given  ax^  +  by^  +  <?2;^  +  2fyz  +  2 ^2:2;  +  2  Arry  =  0 

and  Ix  +  7723/  -f  722;  —  0, 

find  the  condition  that  the  ratios  x  :y  \  z  shall  be 
each  single-valued. 

145.  Eliminate  x,  y,  and  z  from 

f/^  +  9/y  +  h/z  =  0, 
ax  +  Py  +  yz  =  0, 

\ax{s-x)]^+\Py{s~y)\^  +  \yz{s-z)}^  =  0, 
2s  =  x-\-y  -{-z. 

146.  If         u  =  ax''  +  hf+cz^  +  2fyz  +  2gxz  +  2hxy 

be  resolvable  into  linear  factors,  and  if  w  =  0,  then 
will 


a  h  ax-\-liy-\-gz 

h  b  hx-\-by-{-fz 

ax-\-hy-\-gz  hx-{-by-\-fz  0 


0. 


DETERMINANTS. 


385 


147.  Prove  that  the  roots  of 

a  —  X         h  g       =0 

h  h  ~  X        f 

9  I        G-x 

are  all  three  real. 
[See  last  part  of  problem  36  of  this  Exercise.] 

148.  Show  that  the  roots  of 

=  0 


a  —  X 
h 

h 
h~x 

9 
f 

k 

I 

9 
h 

{ 

c  —  X 
m 

m 
0 

lie  between  the  roots  of  the  equation  of  problem 
147. 

149.  Find  the  condition  that  the  equation  of  problem  148 

shall  have  equal  roots. 

150.  Show  that  the  roots  of 

a  —  aiX     h  —  hiX    g  —  g^x    =  0 

h  —  hiX     h  —  hiX    f  — fiX 

9-9i^    f  ~  fi^     c—CiX 
are  real. 
[Reduce  to  a  determinant  of  the  form  of  that  in  prob- 
lem 147.] 

151.  Eeduce 
ax  +  ai  mx  +  mi  Ix  -\-  l^ 
hx  +  hi      hx  -\-  bi  kx  +  ki 
9^+9i  fa+fi  cx  +  ci 

to  a  determinant  of  the  third  order  with  x  in  the 
principal  diagonal  elements  only. 

152.  If  A,  B,  C,  ~F,  G,  -IT denote  the  minors  of 

a     h 
h     b 
9    f 
show  that 


a 

h 

9 

h 

b 

f 

9 

f 

c 

386 


DETERMINANTS. 


ax-\-  hy-\-  gz       hx-{-  hy-{-  fz      qx-\-  fy^  cz 

Ax+Hy+Gz     Hx+By+Fz     Gx+Fy+Cz 

X  y  z 

can  be  resolved  into  linear  factors. 

Find  the  factors  in  the  case 

a=3,  ^--^4,  ^-5,/=l,  ^  =  2,  A=3. 

153.  Show  that  13 o;^  ^by"^  —  l^xy  ~2  is  a  factor  of  the 

resultant  of  the  elimination  of  z  from 

10?/^  + 132^-62/2  =  242 
and  bz^  +  10 x''  - 2xz  =  98. 

154.  Eliminate  y  from 

a^  (x^  +  ^y  +  y^)  —  cixy  (x-\-y)  -{-  x^y'^  ===  0, 
«'(3/'  +  y^  +  2')  -  ayz  (y  +  2)  +  y'^z'  =-  0, 

and  show  that 

d^  {x^  -\-xz-\-  z^)  —  axz  (:r  +  2)  +  x^'^ 

is  a  factor  of  the  resultant. 

155.  If  ai,  a2  are  the  roots  of  a^x^  +  '^^h^x  +  c^  ==-■  0, 

A,  A    ''      ''       "      ^'   a2a;^  +  2Z)2^  +  c,  =  0, 
yi,  72    "      "       "      *'   a3^^  + 2^.3^ +  ^3  =  0, 
form  the  equations  whose  roots  are 

(i.)  2 2^1  =/?iy2  +  y^^yi,    2 2^.,  =  Ayi  +  ^^272, 

(ii.)  2?;i   =  aiy2  +  aayi,      2  '^2  =  ajyi  +  a^y^, 
(iii.)  2tt;i  =  aift+  a^fi^,      2w2  =  a,l3i+  a^/S^, 
and  show  that  the  elements  inverse  to  u,  v,  and  w, 
respectively  in 


1         ^1 

b. 

bi       a^Ci 

axa^w 

b^     aia^w 

a^iCt 

bs     aia^v 

a^a^u 

all  vj 

mish. 

DETERMINANTS.  387 


[The  required  equations  are  the  elements  inverse  to 
Ci,  ^2,  and  C3  in  this  determinant.] 

156.  li  ax^  +  hy"^  -{-  cz^  -\~  2fyz  +  2gxz  +  2  hxy  be  resolva- 

ble into  linear  factors,  the  coefficients  of  y  in  these 
factors  will  be  the  roots  of 

au^  —  2hu  +  b  =  0, 
and  those  of  z  will  be  the  roots  of 

au^  —  2yu  +  ^  =  0, 
and/ must  be  a  root  of 

au^  —  2  hgu  +  h^c  +  g'^b  —  a6c  —  0. 
Example.    For  what  values  of  u  will 

a;'  +  12y'  —  21 2^  -  't^y^  —  4:r^  +  7:ry 
be  the  product  of  linear  factors  ? 

157.  Apply  problem  155  to  determine  the  conditions  that 

ax^  +  by^  +  cz^  +  dw'^  +  2/y2;  +  2^:r2;  +  2  A:?;y 
4"  2  /:rw;  +  2  myit'  +  2  nziv 
shall  be  the  product  of  linear  factors. 
For  what  value  of  n  will 

x'  +  lOy'  +  9^^^  +  5i^'  +  18yz  +  6a;2;  +  6a;y 
+  2xw  —  2yw  +  2nzw 
be  the  product  of  linear  factors  ?    Find  the  factors. 

158.  If         :^'  +  3  a^xhj  +  3  b.xy''  +  c?i?/'  +  3  a.,x'z  +  3  ^2^2' 

+  c^z^  +  3  a3y^2;  +  3  b^yz^  +  ^  exyz 
be  resolvable  into  linear  factors,  determine  the  con- 
dition that  these  factors  should  vanish  simultane- 
ously for  values  of  x,  y,  2;,  other  than  zero. 

159.  If  u  =  ax'  +  by'  +  cz'  +  2/3/2  +  2gxz  +  2 hxy, 

and  2;  =  a^x'  +  b^y'  +  Ciz'  +  2/3/2;  +  2 ^1.^2;  +  2  Ai:r3/, 
find  the   equation   expressing  the    condition   that 


388 


DETERMINANTS. 


u  +  kv  shall  be  the  product  of  linear  factors.  If 
this  equation  have  equal  roots  in  k,  show  that  the 
resultant  of  the  elimination  of  y  between  z^  =  0 
and  ^  =  0  has  equal  roots  in  x/z. 

160.    Find  the  square  root  of  the  resultant  of  the  elimina- 
tion of  u  from 

au^  —  2  XV?  +  a  =  0, 
ait^  —  2yu  —  a  =  0. 

Eliminate  x  from  ax'^ -{-hx^c  =  0  and  x^  =  1. 

Eliminate  x  from  ax^  +  hx^  +  cx"^  +  dx  +  e^=  0  and 
x'^^l. 

Eliminate  x  from 

x'  +  e>Ax'-^Bx+  C=0 
and  2y'  +  2xy''  +  x''y  +  ?>Ay  +  B  =  0, 

and  show  how  to  apply  the  resultant  to  obtain  a 
solution  of  the  quartic  in  x. 

164.    Given  p=h  ~a,  f3i=--  c  —h,    /3.,  =  d—c,    fis^e—d, 
y=l3,~l3,  yi-A-A,  y.-P.-P.. 

S  ==yi— y,    81-=  y2—  yi, 
7/  ==  81—8, 


161. 
162. 

163. 


show  that 

a     ^  y 

P  y  8 

y     8  7; 


165.    Show  that  if 


a  o 

b  c 

c  d 

I 


h  c 
c  d 
d  e 


0,  then  w411 


{ac  —  1)^)  j  ax"^  +  2hxy -f  cy"^    bx^  +  2cxy  +  dy"^ 
\ba?-{-2  cxy  -j-  dy'^   cx^  -j-  2  dxy~\-  e^f 
ax  +  Z>?/     bx  +  ^V 
^o;  +  (?y     ex  -\-  dy 


DETERMINANTS. 


389 


.    Given  c 

XqX^  +  4  aiX^  +  6  a^x"^  +  4  ag^r  +  a^  = 
YoU  +  7i^  +  y^w  =  0, 

-0, 

show  that 

ao      ai      a2 
^0      /?!      A 

7o    yi    7-2 

-27 

ao     «!     as 
«!     ^2     as 
a2     as     a^ 

1 67.    If  Ai  =  a'"  +  yg"^  +  y"^  +  3"^  +  etc.,  in  which  a,  /3,  y,  8, 
etc.  are  the  roots  of 

aox""  +  aio;""^  +  agO;''"^  + +  a,^  =  0, 

show  that 


/^-(-ao-'X 


ai    ao       0 

2  a2    «!       ao 

3  as   a2       ai 


0 
0 

«0 


ma^  a^_i  a^ 
and  that 

-      a   -        ^~  ^^^"^ 

Ix2x3....m 


A^i    1       0      0 

aS,  /Si      2      0 

'^'s     ^2  -^1  3 


^m  '^TO-1  ^m-2  ^/n-3 -^-^1 

168.    If  Sn  =  a''  +  ^'^+c^*  +  etc.,  then  will 
-^(a-bf, 

=  :^(a-by(a~cy(b~c)\ 


s„  >s\ 

=:  ] 

Si   S-: 

So  Si  S-i 

Si  s,  s. 

S,  8, 

s,\ 

jSq   Si  S2  S-s 

S,  jS,  s,  s\ 

S-i  Ss  /S4  s^ 

S-i  Si  S^  Sg 
Generalize. 


=  :^(a-by(a~-cy(a~dy 
(b-cy{b-dy{c-dy. 


390 


DETERMINANTS. 


169.    If  ai,  ^2,  <^3>  <^n  ^^Q  the  roots  of  f(xY=^  0,  show 


that 


K^r- 


X  «!  ai  ai 

ai  X  a^  a^ 

a.^  a^  X  a-s 

as  as  as  ^ 


an 


a^ 


an 


a^ 


ai 

ai 

^2 

a., 

^3 

as 

a^ 

a, 

X 

an 

c 

c 

170.    If  B^ 


'  +  ar  +  a-r  + +  ar,  show  that 


1        X     x"^ 

Oq  ^1      ^n 


0„_i    bn     btn-\ 


X 


X 

ai 

a, 

ai 

X 

a^ 

a, 

a, 

X 

171.    Resolve  into  factors, 


-^0  Ol    bn~\ 

Si  S2  Sn 

S\  Ss 


cy  o 


ai 

«i 

a^ 

^2 

as 

as 

X 

«n 

1 

1 

0  1 

1  8o 
y    8, 
t    8. 
f    8, 

8,    8,    8, 
8,    8,    8, 
8,    8,    8, 
8\    8,    8, 

in  which  8,^  i 

=  a''+b"  +  c" 

.    Show  that 

8,    1 

8,   8, 
8,    8., 

0         0 
2        0 
8i       3 

...      0 
...      0 
...      0 

8,.  8„. 

1  &L-,  isL. 

...   m— 1 

...      8, 

8„. 


b'ln 


DETERMINANTS. 


391 


-(-i)nx2x3 

but  =  0,  if  m  >  ??, 

in  which  Sr  =  a/  +  az""  +  a/  + 

r  being  any  positive  integer. 


(fi  —  l)naia2 a„,  ii  771  =  71, 


173 .    If  u^ ■=  aitti"*  +  ^202""  +  asttg"*  +  a^a^""  +  c(5a-J^,  prove  that 

=  0. 


Uo 

Ui 

u. 

Us 

Ui 

U2 

Us 

u. 

u. 

U, 

u^ 

u, 

21, 

u. 

?/5 

u^ 

U, 

U-o 

Ue 

u, 

W'5 

Wfi 

u. 

u^ 

u. 

u. 

u. 

Ue 

Ue 

U^ 

^7 

u. 

Us 

u. 

U9 

Uio 

174.  Obtain    G   in    determinant   form   by    eliminating   x 

between 

and  ax  -{-  b  —  1/  —  0. 
(See  §  62,  p.  297.) 

175.  Obtain    I   in    determinant   form    by    eliminating   x 

between 

ax'  +  Abx'  +  6cx^  +  4:dx  +  e=^0 
and  ax  -\~b  -~  'i/  =  0. 
(See  Ex.  70,  prob.  1,  p.  312.) 

176.  Express  IT,  A,  /,  J,  and  P  — 27  J^Mn  determinant 

form,  given  as  data  the  propositions  stated  in  prob. 
26,  Ex.  68,  and  probs.  14  and  15,  Ex.  70. 

177.  Express  Us,  Is,  Is,  G^s  of  the  equation 

Sox'+4:S,x'+6S,x'+AS.sX+jS,=:^,Xx+ny^O 
in  terms  of  H,  /,  J",  G  of  the  equation 

aox'  +  4  ttiX^  +  6  (22^^  +  4  a^x  +  a^  =  0, 
of  which  Ti,  r2,  r^,  r^  are  the  roots. 


392 


DETERMINANTS. 


178.  Express  Hs,  Zs,  Js,  (^s,  and  //  — 277"/,  as  functions 

of  tlie  differences  of  the  roots  of  the  quartic. 

179.  Express  A^  as  a  function  of  the  differences  of  the 

roots  of  the  cubic. 

180.  If         x  =  XiU  +  /xi'y, 

y  =  X^u  +  iL^v, 
transforms 

ax^  +  2  hxy  +  hy'^ 
mto  Au'  +  2Suv  +  Bv\ 
find  the  value  of 


J,    H 
H,   B 


181.    If 


y  ^\u-\-  ii.p  +  v^w, 

transforms 

ax"  +  by''  +  cz^  +  2fyz  +  2gxz  +  2hxy 
into  Av?  +  Bv"  +  Cw'  +  2  i^i;i^  +  2  ff i^z^;  +  2  Hm, 
find  the  value  of 


A    H  0 

-^ 

a     h    g 

H  B    F 

h     b    f 

G    F    C 

9    f    c 

182.    If 


^  =  {Ky  +  i^iz)/{X,y  +  ii^,z), 
ayX^  +  2hiX  -\-  Ci 

=  (Ay  +  2  B,7jz  +  C,z^)/(Ky  +  /.,.)^ 
and  a2^^  + 2  ^2^4-^2 

=  (A,f  +  2  ^23/^  +  C,z')l{\,y  +  ^2^)^ 
then  will 

^1 2A   a  0 

0     ^1    2^1  a 

^2  2^2  a  0 
0  ^2  2^2  a 


«!  25i    <?i    0 

X 

Ai  /Ai 

0     ai    2^)1  c'l 

^2     /^2 

a^  2h.j^    c-i    0 

0     a^    2Z>2  ^2 

DETERMINANTS.  393 


183.  If  the  quartic  (a,  h,  c,  d,  e){x,  1)*  =  0  be  transformed 

by  the  homographic  transformation 

^  =  (Ky  +  /^i)/(\y  +  /^2), 

then  will 

in  which  J/=    \   /xi 

[Jf  is  called  the  modulus  of  the  transformation.] 

184.  Find  Ay/A^.  for  the  homographic  transformation  of 

the  cubic  {a,b,c,d)  {x,  Vf  —  0. 

185.  If         K-{p'-q')/{p'+q%   f^i  =  2pq/(f+q'l 

A,  —  2pq/{f  +  q'),       /.,  =  (/-  qy(f+  q'\ 
X  ~  XiU  +  /xif ,  y  =  X^^ + /^2'i^, 

then  will  x'^  +  y^  =  u^  +  v^, 

[A  transformation  that  changes  x^-\- a;./+  ^^3^+ +  x,^^ 

into  u^  +  u,^  +  u^  + +  u^  is  termed  an  orthog- 
onal transformation  of  the  nth  order.] 

186.  Form  an  orthogonal  transformation  of  the  third  order 

and  determine  the  value  of  its  modulus. 
(See  prob.  181  above.) 

187.  Form   an   orthogonal   tranformation   of    the   fourth 

order. 

188.  Show  that  H,  I,  J",  G,  P-21J^  ^yq  the  same  for 

both  the  quartics 

a^^ + 2  h(ficy  +  c^y"^     aiX^ + 2  h^xy  +  c^y^   =  0, 
aiX^  +  2  bixy  +  c^y"^     a^x'^  +  2  h^xy  -j-  c^iy"^ 

Gqx'^  +  2ain;3/  +  ^2^^     ^0^^  +  2  ^i.t?/  +  h.^y"^  I  —  0. 
box'^  +  2  Z^i^ry  +  b^y"^     CqX^  +  2  Cjo:?/  +  <?2y^  I 


394 


DETERMINANTS. 


189.  Apply  Example  7,  p.  336,  to  solve  the  cubic 

190.  Form  the  equation  whose  roots  are  the  products  in 

pairs  of  those  of  x^  +  px^  +  ^:r  +  r  —  0. 
[ai  =  /?y,  .*.  aai  =  aj8y==  — r.  .'.a^y  +  r^O;  eliminate  :i\] 

191.  Form  the  equation  whose  roots  are  the  products  in 

pairs  of  those  of  x^  -\-px^  +  qx^  -\-rx-\~  s  —  O. 

192.  a,  p,  y  being  the  roots  of  the  cubic  (a,  b,  c,  d)  (x,  If  —  0, 

form  the  equation  whose  roots  are  a^  +  y,  ySy  +  a, 
ya  +  yS. 


193.    If  a,  (3,  y  be  the  roots  of 


-0, 


then  will  py,  ay,  ap  be  the  roots  of 


X 

0 

0 

—  a 

1 

X 

0 

h 

0 

1 

X 

~  c 

0 

0 

1 

d 

'^000 

y   0   1 


1 

0 
0  10  0 
abed 
b     c     d    0 


:0. 


194.    U{a,b,c,d){x,yy  =  A{x  +  e,yy  +  B{x  +  e,y)\  show 
that  ^1,  02  are  the  roots  of 
0. 


a 

h 

c 

b 

c 

d 

1 

e 

0" 

195.    If  the  cubic  (a,  b,  c,  d){x,  Yf  =  0  be  transformed  into 
a  cubic  in  y  by  means  of  the  equation 

y  =  a{ax  +  b)  +  P^ax"  +  ^bx  +  2c), 
show  that  this  cubic  is 


DETERMINANTS.  395 


--ab-~2f3c  aa-S/3b  —pa 

pd  y  —  ab-\-     Pc         —ab 
ad  3ac+     pd  y  +  2ab  +  pc 

196.  Determine  the  condition  that  the  roots  of 

{a,b,c,d){x,Yf  =  Q 
may  be  formed  from  those  of 

(?7i,  n,  p,  q)  (x,  ly  --  0 
by  adding  the  same  quantity  to  each. 

197.  Given  ax^  +  3bx'' +  ocx  +  d  =  0 

express  in  the  form  of  a  determinant  equated  to 
zero,  the  equation  whose  roots  are  Ix  +  '^^y. 

198.  Being  given  the  cubic  (a,  b,  c,  d)(x,  ly  —  0,  express 

ni,  p,  and  q  in  terms  of  a,  b,  c,  and  d,  so  that  the 
vahies  which  7717/'^  -{-  2  ^yy  +  q  takes  when  y  is 
replaced  successively  by  a,  P,  y,  the  three  roots  of 
the  cubic,  are  the  three  roots  in  the  order  p,  y,  a. 

199.  Determine  the  relation  that  must  exist  among  the 

coefficients  of  the  cubic  (a,b,c,d)  (x,  1)^  —  0,  in 
order  that 

Aa+Bp+Cy  =  0, 
a,  p,  y  being  the  roots  of  the  cubic. 

200.  If  a,  p,  y,  aj,  pi,  yi,  are  the  roots  of  the  cubics 

(a,  b,  c,d)(x,iy  =  0,     (ai,bi,Ci,d)(x,iy  =  0, 
form  the  equation  whose  roots  are  aai  +  ^^1  +  yyi, 
etc. 

201.  a,  p,  y,  S  being  the  roots  of  a  quartic,  form  the  equa- 

tion whose  roots  are 

PyS  +  yS  +  pS  +  Py  +  p  +  y  +  S,  etc. 


396  DETERMINANTS. 


202.  Also  the  equation  whose  roots  are 

(a-fi)(a-y){a-8),((3-y)(^-S)(p-a), 

203.  And  the  equation  whose  roots  are 

(a-/3)(y-8),   (a-y)(8-^),   (a-8)(P~y}. 
From  this  result  prove  that  if  I^  —  27J'^  =  0,  the 
quartic  will  have  equal  roots. 
(Seeprob.  14,  Ex.  70,  p.  312.) 

204.  If  a,  ^,  y,  8  be  the  roots  of  {a,b,c,  d,e)(x,iy  =  0, 

then  will   the  equation  whose  roots  are  (a  —  /3y\ 
(a  —  y)^,  etc.,  be 

-3  a  a'z  la'z'+m+4:I=0. 

ah  a{\ah'+^m+I)  6J" 

\  aV+4  Hz+I  alz+^  J  --2Jz 

205.  a,  /?,  y,  S  being  the  roots  of  the  quartic 

(a,  5,  c,  d,  e)  {x,  1)*  =0, 
express  the  product 

\x-{l3-yy}\x-{a-m 
in  terms  of  a,  h,  c,  d,  and  a  single  root  of  the  re- 
ducing cubic  f  -~  It  +  2J=^  0,  and  hence  form  the 
equation  of  the  squares  of  the  differences  of  the 
quartic. 

206.  Solve     \x-a{al^+y^)]\x-a{ay+P^)\{x-a{a^+Py)} 


207. 


==15 

a     b     c 
b     c     d 
c     d    e 

in  which  a,  ^,  y,  8  are  the  roots  of 
{a,h,c,d,e){x,iy=^0. 

Find  the  relat 
ibhe  cubics 

U={a, 

ion  which 

b,  c,  d) 
b\  c\  d^y 

connects  the  coefficients  of 

(^,  1)', 
{X,  l)^ 

DETERMINANTS. 


397 


when  it  is  possible  to  determine  the  ratio  X//x  so 
that  XU -{-  fjiV  msij  be  a  perfect  cube. 

208.  If  the  roots  a,  /3,  y,  S  of  (a,  h,  c,  d,  e){x,  Vf  =  0  are 
all  unequal,  and  if  there  exist  unequal  magnitudes 
0  and  ^,  such  that 

(a+6)*:(/3+6y:(y+^y:(8+6)* 
::(a  +  ,^y:(^+<^/:(y  +  <^)*:(8  +  <^y, 


show  that 


a 

b    c 

h 

c     d 

c 

d    e 

=  0, 


and  form  the  quadratic  determining  0  and  <^. 

209.    If  _J_  +  _2_+JL=0, 

a  —  p      y — a      a  —  o 

in  which  a,  /?,  y,  8  are  the  roots  of 

(a,  h,  c,  d,  e)  (x,  1)*  =  0, 


show  that 


a  b  c  \ 
be  d\ 
c     del 


0. 


210.  If  a—  -y/^,  a,  and  a+  -^yS  be  three  of  the  roots  of 

the  quartic  (a,  b,  c,  d,  e)  {x,  1)*  =  0,  show  that 

5,         0,  3^  ~G,  0  -0. 

0,        5,  0,  3^,  -G 

27^,  21(9,  b(a'I-Sir'),  0,  0 

0,  27^^,         21(7,         ^ia'T-SIT'),  0 

0,         0,  27^,  21(7,         5(a'I-Sir') 

211.  Show  how  to  solve  the  quartic  (a,  ^,  c,  d,  e){x,  1)*=  0 

by  assuming 

ax'^  +  ^bx^  +  ^  ex^  +  Aidx  +  e 

=  a  (:r^  +  2z^i:r  +  V])  {x^  +  2  u^x  + 1^2) 

'tvjt/'2  —  e      c, 
and  eliminating  t^j,  u^^  Vi,  -^x- 


398 


DETERMINANTS. 


[110 

1     Wi    V.J    = 

Ux  u-i  0 

1  iCi  ?;, 

t^i  v^i  0 

0  0    0  i 

a          h      c  +  2t 

b       c  —  t        d 

c  +  2t 

d          e       1 

212. 


213. 


ih  +  ii-i  Vrri'i 

Hi  +  ?/.,         1^  UiU.,         U^V^  +  U.^Vi 

:0;  i.e.,  4^^-7^  +  or=0.] 


Show  that  tlie  reduction  of  the  quartic 

{a,b,c,d,e)(x,iy^O 
to  the  biquadratic  form 

{x"  +  1)^  +  Ax  {x'  +  1)  +  Bx''  =  0 

depends  upon  the  solution  of  the  cubic 

(ax'  +  4:bx^  +  6cx^  +  4:dx  +  e)  {ax  +  by 
-=  a{ax^  +  Ux'  +  ?>cx  +  d)\ 

If  a,  p,  and  ttj,  ySi  are  the  roots  of 

o,x'^  -\- 2bx -{-  c '-=  0  and  a^x^  +  2 ^i.r  +  <?i  =  0, 
respectively,  show  that 


a 

2^ 

c 

0 

0 

a 

25 

c 

a, 

2^^, 

^1 

0 

0 

ai 

2^1 

^1 

214. 


215. 


Similarly,  resolve  the  resultant  of 

ax^+Zbx'+2>cx  +  d=0  and  a^x''+2b^x  +  c^-=0 
into  a  product  of  the  differences  of  the  roots  of  the 
two  equations. 

By  eliminating  g,  A,  Jc,  and  /  from 

a  =  {ga  +  h){]ca-Vl),     ^  =^  (gb  +  h)(B  + 1), 
y^(gc+h){kc  +  l),     S^(gd+h)(kd+l), 


DETERMINANTS. 


399 


=  1  ah+cd  a/3  +  y8 
1  be  -{-  da  Py  +  8a 
1     ac-\-bd    ay  +  pS 


prove  that 

1  a  a  aa 

I  h  P  bp 

\  C  y  cy 

1  d  S  d8 

^(a-/3Xb-cXy-S)(d-a) 
~{a~b)(P-y){c~d){8~a) 

=  (a-y)(c-d)(S~P){b~a) 
-(a-c)(y-8)(d-b)(l3~a) 

^(a~-8)(d-b)(/3-y)(c-a) 
-(a-~d)(8-l3)(b--c)(y-a). 

216.    Similarly,  prove  that 


1     a  aa 

1     b  bp 

1      c  cy 

and  that 


=(^a-b){^-y)c-{a-^){b-c)a, 


1  a-\-  a  aa 
1  b+p  b/3 
1     c  -{-  y     cy 

^(a-b){l3-y)(c-a)  +  io^-(3)(b-c)(y-a). 

217.    a,  (3,  y,  8  and  ai,  /3i,  yi,  8,  being  the  roots  of  two 
quartics,  prove  that  if  T^Ji^  =  Ii^J^\  then  also  will 

:0. 


1 

a 

cti 

tttt] 

1 

P 

/8. 

/8^. 

i 

y 

yi 

rri 

1 

8 

88, 

218.  If  u,  V,  w  denote  the  roots  of  aV  —  aIx-{-2J~0, 
and  Ui,  Vi,  lUi  the  roots  of  ajV—  aiIiX~{-  2  Ji  —  0, 
a,  p,  y,  8  the  roots,  and  /  and  JT  the  invariants,  of 
(a,  ^,  c,  d,  e)  (rr,  1)*==0,  and  ai,  ^i,  yi,  8i  the  roots,  and 
Ii  and  f/i  the  invariants,  of  (ai,  &i,  Ci,  c?i,  ^i)  (a;,  1)*  —  0, 
then  will 


400 


DETERMINANTS. 


1  a  aj  aai 

1  /5  A  PP. 

1  y  yi  yyi 

1  8  8i  8Si 


-4jl 
1 
1 


u 

Ui 

V 

Vi 

10 

IV, 

a 

/8i 

tttti 

I 

I 

8S, 

219.  If  a,  y8,  y,  S  be  the  roots  of  (a,  5,  e,  c?,  e)  {x,  ly  =  0, 

and  ttj,  /?i,  yi,  8i  those  of  (ai,  ^i,  Cj,  Jj,  Ci)  (:r,  1)*  =  0, 
form  the  equation  whose  roots  are  the  twelve  dif- 
ferent values  of 

1 
1 
1 
1 

220.  If  (a  -  aO  (/?  -  ^0  +  (a^  -  ^)  (^  -  a)  -  0, 

in  which  a  and  p  are  the  roots  of  ax^  +  2  Z>a;  +  c= 0, . 
and  ttj  and  Pi  are  the  roots  of  aiX^  +  2h^x-\-Ci  —  0, 
show  that 

aci  —  2hh,-\-  cai  =  0, 
and  that  aci  —  2hhi-\-  cai  is  a  factor  of  the  invari- 
ant J  of  the  quartic 

(ax'  +  2bx+c)  (a.x'  +  2  b,x  +  c,)  =  0. 

221.  Reduce 

\x^  +  y'-(l-Jc)gx  +  7n''\' 
=  il  +  Jcy\/i^(x'  +  f)-gyi 
to  the  form 

\x''  +  f  +  Ax  +  B\'  =  ax''  +  hx  +  c, 
and  show  that  h'' —  ^ABh +  ^A'c  =  0. 

222.  So  determine  h  and  I  in  terms  of  a,  h,  and  o,  that 

(x''+  7/'+  zy+  2  ax'+  2  h7/'+  2  cz'+  2  Jcx+ 1 = 0 
may  for  y  —  0  assume  the  form 

(x'  +  z'  +  7n){x'  +  z'  +  n)--=0; 
and  for  z  —  0,  the  form 


/? 


DETERMINANTS.  401 


■     223.    If  a,  p,  y,  8,  four  of  the  roots  of  the  quintic 
{a,h,c,d,ej){x,  1)' -- 0, 
be  connected  by  the  relation  a  +  yS  ==  y  +  8,  show- 
that  €,  the  fifth  root,  will  be  given  by  the  equations 

224.  If  a,  /?,  y,  8,  four  of  the  roots  of  the  quintic 
(a,  h,  c,  d,  ej)  (x,  If  =  0, 

be  connected  by  the  relation 

S(2a-^-y)(2^-y-a)(2y-a-/3)  =  0, 
then  will  €,  the  fifth  root,  be  determined  by 

z=^ae  +  b,     z'  -SSz+lQG^O. 

225.  If  (a-^)(y-8)  +  (^~y)(8-a)-0,   in  which  a,  ^, 
y,  8  are  four  of  the  roots  of  the  quintic 

asf  +  b  =  c{x+Vf, 
then  will 

j^^  -  2' 3  VJT  +  2^^  3'iy  =  0, 
in  which 

J  =  b'c'  +  cV  +  a'b'  -2abc(a  +  b  +  c), 

K=  a^b'^c'  (be  +  ca  +  ab), 

L  =  a'b'c\ 

226.  Reduce  (a,  b,  c,d,e,f)  (x,  yf  to  the  form 
K  (^  +  Kyf  +  ^2  {pc  +  k{yf  +  ^3  (^  +  %)^ 

and  hence  prove  that  ( )  (a;,  1)^  =  0  can   be  re- 
duced to  the  form 

l{x^  Xf  —  mx^  —  n  =  0, 

into  linear  factors. 


.    Kesolve 

0 

1 

1 

1 

1 

0 

c^ 

b' 

1 

c' 

0 

a' 

1 

b' 

a' 

0 

402 


DETERMINANTS. 


228. 


,   P,    y  being 
express 


the    roots    of 


-  2)X^  +  qx  —  r  =  0, 


0 

a 

/? 

y 

-4- 

0     1 

1 

1 

y 

0 

y 

y^ 

1     0 

y 

/^ 

fi 

y 

u 

a 

1    y 

0 

a 

a 

/^ 

a 

0 

i    (^ 

a 

0 

in  terms  of  ^9,  5',  and 


229.    Find  the  value  of 


230. 


0 

a 

b 

a 

0 

c 

b 

c 

0 

c 

b 

a 

Show  that 

0 

a^ 

h 

a' 

0 

Y 

b' 

/ 

0 

& 

^' 

a' 

0 

1 

1 

1 

1 

0 

a 

^ 

1 

a 

0 

Z> 

1 

b 

^ 

0 

1 

c 

c 

a 

231.    If 


0 

=  16^^  (;?^  -  aa)  {f  -  bp)  (p'  -  cy) 
2^'  =  aa  +  b/3  +  cy. 

^0, 


0     1 

1      1 

1 

1     0 

a'      /3-^ 

t 

1      a^ 

0      c' 

1     fi' 

c'     0' 

a' 

1     / 

V     a? 

0 

then 

will 

(a^  +  ¥  +  c^  +  a^  +  p''  +  y'^)(aV  +  b^^P"  +  c^^) 
=  2  a^a\a'+  a")  +  2  ^^^9X6*^+  /?^)+2  c  V(c^+  y') 
+  a^)8V  +  bya^  +  c'^a^yS*^  +  a'b''c\ 

232.    If  a'+/?'  +  a^  =  C^ 

then  will 


DETERMINANTS. 

4(J5 

0      1 

1 

1 

1 

-0. 

1      0 

a' 

P' 

y'^ 

1       a' 

0 

c' 

^^ 

1     1^' 

c' 

0 

a' 

1      / 

h' 

a' 

0 

233. 

If          x^  +  if 

-2 

i.xy  = 

-^^ 

and 

1    r 

li 

=  0, 

f  +  z^- 

~2pyz  = 

=  «^ 

y    1 

a 
1 

Z^  +X'' 

-  2yzx  - 

=  ^', 

P     a 

then  will 

0     1 

1 

1 

1 

=  0. 

1     0 

x'' 

y' 

^2 

1     x' 

0 

c*^ 

^'^ 

1     f 

c' 

0 

a^ 

1      z^ 

h' 

OJ" 

0 

234. 

Prove  that 

0 

X 

y 

z 

;  =  (a:r  - 

-by  + 

cz)\ 

—  X 

0 

c 

b 

~y   - 

—  c 

0 

a 

■—  z     - 

-h 

—  a 

0 

and  generalize  the  theorem. 
235.    Evaluate 


X 

y    z 

y 

X     y 

z 

Z       X 

z     and  use  the  result  to  prove  that 


u^  -\~  v^  -\-  w^  —  ?>  uvw 

=  {a^  +  h^  +  c^-?> ahc) {x^  +  f  +  z^-  Zxyz), 
wherein 

u  =  ax-\-by-^cz,  v^=cx-\-ai/-\-hz,  w  =  bx-{-cy-\-az. 

236.    Evaluate 

and  use  the  result  to  prove  that 

1  -  (2x'  -  ly  ~  (2y'-iy  -  {2z~Vf 
+  2{2x'  ~l){2f  -l){2z'  -V) 
=  -4c(l-x^-y''-z'+2xyz){l~x''-y''-z'-2xyz). 


1 

X 

y 

X 

1 

z 

y 

z 

1 

404 


DETERMINANTS. 


237. 


-  c 
-d 


h 
a 
d 


a 
b 


d 

c 

-b 

a 


a  +  ib 
—  c-\-id 


■  id 
id 


=  (a'  +  b'  +  c'  +  dy,  i'  +  l  =  0. 
Apply  this  identity  to  prove  tliat  the  product  of  the 
sum  of  four  squares  by  the  sum  of  four  squares  can 
be  reduced  to  the  sum  of  four  squares. 

238.    Prove  that       a  pb        qc  pqd 

—  b  a  —qdj       qc 

—  c  pd        a  ~pb 
— d  —c        b  a 

=  (a^  +^&^  +  qc^  -\-pqdy. 

Hence  prove  that 

(a^^pb''-\-qc^-^pqd''){A^^pB^\qC''-^pqB') 

=  (a  A  +pbB  +  qcC+pqdny 
+p  (-  aB  +  bA-  qcB  +  qdC)' 
+  q{-  aC+pbD  +  cA~ pdBy 
+pq(-  aD  -  bC+  cB  +  dA)\ 


239. 

Show  that 

a^         "lab         b' 
aa     ap  +  ab     bp 
a'           2  a/?          p-" 

=  {ap  +  ab)\ 

and  generalize. 

If  u  =  {x--  ai)  {x  —  a.2)  (x  —  aa) (x  ~  a„) 

=  x""  -^1^;"-^  +i?2^'"~' i~TPn,  show  that 

240. 

0     1111    

1     X     a^ 
\     ai     X 

1        «!        «2 

1     «!     a.j_ 

Cla     a^  

«3        Cii     

X     a^  

^3     X    

---nx''-^  +  {n—l)p^x''-''-{n-2)p.,x''-^+ 


DETERMINANTS. 


405 


241. 


242. 


0  111 

1  ai  X  X 
1  X  a^  X 
1.  X       X       ct^ 


=-(  —  iyinx''-'^-(n~l)p,x''-''+ \. 


X      a^     a^ 
Oi     X      as 

(X\       C(/2       X 


=  w  +  S 


X  —  Gr 


243.    Show  that 


a-}-  X  X  X 

X  P  -{-  X  X 

X  X        y-{~  X 


X 

X 
X 


X  X  X         h-\-  X 

=  apyZx{/a  +/^  +/y  +/8  +/x). 


"ove  that 

0 

1 

1 

c 

1 

a 

1 

a 

1 
a 

-(o- 

a  —  b 

-by 

1 
1 
1 

b 
b 
b 

c 

b 
b 

a 
c 
b 

a 
a 
c 

and  gener 

alize. 

245.    Writing  /(:r)  for  (ci~x)(a2  —  x) (Cn  —  x),    show 

that 

Ci     a     a  a    _af(b)  —  bf{a) 

b      c^     a  a    ~         a  —  b 

b      b     Cj,  a    ... 

b      b     b  Ca    ... 


246.  Show  that  0  a^  a^  a^  % 
^1  0  a^  a^  a^ 
bi     b^,     0      a^     ^5 

^1       ^2       ^3       0        % 

bi     b.2     Z>3     b^     0 
=  bia^a^a^a-^  +  b^h^a^a^a^ + bf^^Jj^a^ar,  +  b^^Jj^a^. 


406 


DETERMINANTS. 


247. 


248. 


249. 


250. 


a  b 

c  h 

a  a 

h  h 

c  c 


h  h  h  h 

c  c  c  c 

a  a  a  a 

a  b  h  b 

c  b  c  c 


a     a     a     a     a     c     a 
b     b     b     b     b     b     a 


\(ia-b){b~c){c~a)\'' 


X     1 


-  +  - 


c — a      a~b 


c 
b^c 


1  1 

a  b 

c  b 

a  a 

b  b 


c     c     c 
a     a     a     a 


a  a 
a  b 
c     b 


=  n\{a-b){b~c){c-a)\''-'' 
X  {a'  +b''+c'~bc  ~  ca  -  ab). 

Prove  the  two  following   identities,  and   generalize 
them  : 


a  —  X        b  c 

b         c  ~~  X        a 
c  a        b--x 


■-.{x~B,){x'-SA\ 


/S'„  =  a  +  a>"Z>  +  a>'"c,      0)2  +  0)  +  1  ^  0. 


a  —  X      b  c  d 

b      c  —  X      d         a 
c  d     a—x      b 

dab  c  —  X 


=  {x-8,){x-8,){x'-8A), 


8n  =  a  +  i^'b  +  i'^'c  +  i'%     i' +  1  =  0. 


DETERMINANTS. 


407 


251.  (S-uy  x'  f  z" 
u^  {8-xy  y"  z^ 
u^  x^  iS-yf  z^ 
u"             x"             y"        (S-zf 

=  2  S'uxyz  (/u  +/x  +/y  +/z  -  4:/S) 
wherein 

S=  u -{-  X -{- y  -\~  z. 

252.  0         1  1  1 

1  Q/+zy     f 

1        x^       (x  +  zj        z" 

1      x"         y'     {x+yy 

~{x  +  y  +  zyiix''+y''  +  z^-2yz-2xz~2xy). 

253.  0         1111 

1  {8-uy        x'  f  z" 

1     1^     "     "     ' " 

1        y? 
1        u' 


(s-xy     r 
x'     (S-7jy     z' 
x'        f    (s~zy 

^S'\uXS~2u)+x\S-2x)  +  2/(S-2y) 
+  zXS-2z)+2ux7/z(/u+/x+/y+/z)l 
S=u  +  x  +  y-{-z. 
Prove  the  four  identities  next  following,  in  which 
U=  (x  -  2ai)  (x  ~  2a,) (x  -  2a„). 


254. 


(x  ~-  a  J 


a.' 


as 
a^'  (x  —  tta)^  ai^ 
ai  a^       {x  —  a^y 


255. 


I  x  —  2a^) 


0  1  1              1 

1  {x  —  a^'^       a^i  ai 
1  a^        {x  —  a^^        a^ 

1  a^            a^'       {x~a-^^ 


=  —  nUx'"-  ^  5 


\ 

^  X~2ara 


408 


DETEEMTNANTS. 


256. 


257. 


I  x~2a^) 


0         «!  a^  as 

ai  {x—a^y       a^a^  o^^a^ 

a^       a^a^      {x  —  a^       a^a^ 


=  -  ^ri;*^-^  S 


a;  — 2a^ 


258. 


X  —  Ai       a2  as 

a\      X     -^2       ^3 

(Xj  (X2         ^        -^3 


=a;(^-/^)--\ 


259.    If  /(a,  ^>)  = 


«!  (22  <^3        o:  — ^ 

y(S'=ai  +  a2  + +  ««,     -4m  =  >S'— a, 


I  ,  a  —  "TYi  ,  a~n 


b  —  k      b  ~  I      b  ~  7)1 
show  that 

/(a,  a)    /(5,a)    /(.,  a)    f{d^  a)  ^  0. 
/(a,  5)    /(5,5)    /(.,^)    f{d,b) 
f(a,  c)    f(b,  c)    f(c,  c)    f(d,  c) 
f(a,d)    f(b,d)    f(c,d)    f(d,d) 


260.    Expand 


261.    Expand 


A  —  c  2a  Sa  2a 

2b  X  —  c  2a  Sa 

Sb  2b  X-~c  2a 

2b  Zb  2b  \-c 


a 

b 

c 

d+X 


b  c       d —  X 

c       d+^X       e 
e  f 

f  9 


d~\X 


DETEEMINANTS. 


409 


262.    Show  that  if 

(x  —  a){x  —  b)  (x  —  c)  ^x^- 
a  +  b  —  c        4a  6a 


•px 


6b 
45 


a  +  b- 

4:b 

6b 


c        4a 

a  +  b- 

4:b 


qx- 

4:a 
6a 
c        4:a 
a-{-b  - 


■r, 


p  q        r     V 

—  4:  p        q     r 

0  —16    p     q 

0  0      -4:  p 


263.  If  a-\-b-\-c-—0,  ab-\-bc-\-ca—  q,  abc~r,  then  will 

X-h'c'  ab(a'  +  c')  ac(a'  +  b') 
ab  (b'  +  c')  X  -  aV  be  (a'  +  b') 
aclb'  +  c')     bcia'  +  c')        k-a'b' 

=  \'-Xq  (q'  -  2r')  +  q\ 

264.  Show  that 

X-~2bp  —  2cy        aP  -}-  ab  ay-\-ac       I 

a^  -{-  ab        \  —  2aa  —  2cy         by-{-  jSc 
ay+ac  by  +  ^c        \—2aa-2bp\ 

=  [(X  +  aa+bp  +  cyy 

-{a'+b''+c'){a^+p''+y'')\(X-2aa-2bl^-2cy). 


265.    Show  that 

X  +  aa  +  a'a' 
ap  +  a'yS' 
ay  +  a'y' 


5a  +  Z>V  m  +  cV 

X  +  b/3  +  b'p'     cf3  +  c'l3' 
by+b'y'       X+cy  +  c'y' 


=  X--'{X''  +  X(A  +  ^)  +  AB-CDl  ; 
wherein  n  is  the  order  of  the  determinant,  and 

A  =  aa'  +  bb'  +  cc'  + 

■B  =  aa'+PI3'+yy+ 

C  =  aa^  +  bf3'+cy'+ 

I)=aa'  +  pb'+yc'+ 


410 


DETERMINANTS. 


266. 


267. 


268. 


269. 


270. 


Show  that 

1   a    a     a^ 

I  IS  p  p^ 

1   y   y'  yy' 
1   8    8'   8S' 


1   a  I 

1    /^l 


1  1  1 

a  +  yS  y+8'   S  +  y' 
ap         yS'         By' 


and  generalize  the  proposition. 
Show  that 


a, 

h, 

Cy 

d, 

a. 

h 

C2 

d. 

a^ 

h 

0 

0 

a^ 

K 

0 

0 

^  i  a.,  hAx\  <?i  c?2 


and  generalize  the  proposition. 

Given  {x^  —  x^y  +  {y^  —  y,)'  =  a\ 
(x,  —  x,y  +  (y2  -  ys)'  =  b\ 

(x^  —  x,y  +  {y,-7j^y  =  c\ 

(x,  —  x^y  +  (y^  -  yi)'  =  d\ 

(xi-x,y  +  (7ji-y,y  =  h\ 
(x,-~x,y  +  (i/,-y,y  =  k\ 

j8  = 


1 

1 

«i    2/i 

0 

1 

a-2    y2 

1 

1 

a^s     ys 

0 

1 

a;*     yt 

then  will  16  /^^  -=  4  A^;^^  -  (a'  -h'  +  c'-  dj. 
If  hh=^ac  +  hd  and  2s  =  a  +  ^  +  ^  +  ^^, 
then  will  8^=^{s-  a)  (5  --h){s-  c)  (s  -  c^). 

:0 


ai 

^i 

Ci 

Xi 

yi 

2^1 

a2 

b. 

C2 

^2 

y2 

2^2 

«3 

b. 

C3 

X, 

ys 

23 

is  satisfied  by  any  one  of  twelve  systems  of  three 
equations  each  ;  find  them.  . 

Given  x^^  +  3/1'  -|-  Zi^  =  l,       x.x^  +  y,y,  +  2:12:2  ^  0, 

^•/  +  y-/  +  Z2  =  1,      ^2^3  +  ^2^3  +  2^2^:3  ="  0, 
X'l  +  yl  +  zi  ==  1 ,       x^,x^  +  3/33/1  +  032:1  ===  0, 


DETERMINANTS. 


411 


271. 


272. 


273. 


prove  that 

^1  ~r  ^2  ~r  ^3  "^  -Lj 

and  if 

A  =    x^ 

X, 

X. 

-X 


3/:^ 


^\yi  +  ^22/2  +  ^32/3  =-  0, 

2/i2;i  +  3/2^:2  +  2/32;3  =  0, 

2;i:ri  +  z-,x.2  +  03:1^3  =  0, 
then  will  X  =  ±  1, 


and 


2/1 
2/2  —  '^ 


Z^S--X 


0. 


a 

X 

c 


X 

c 

c 

d     d    y 

d     y     h 

y     h     b 


h 
h 

y 

X 

a 
a 


b 

d 

i 

c 

c 

X 

c 

X 

a 

X 

x  —  y  a  —  h  a~b 

c  —  d  x  —  y  a  —  b 

c  —  d  c  —  d  X  —  y 

x-\-y  a-\'b  a-\-b 

c  ■\-d  x-^y  a-\-b 

c  ^d  c  -\-  d  c  -\-d 


a  p 
a  p 
X    p 

q    2r 


d  d  y  p 
d  y  b  p 
y     b     b     p 


b 
b 

y 
<1 


b 

y 

y 

d 

d 

d 

^ 

<1 

c 

c 

X 

c 

a 

X 

—  ''^'X—y    a  —  b    a  —  b 
c  —  d   x  —  y    a  —  b 

y 


- d    c  —  d    X- 


X 


r       p  p  p 

q  x-\-y  a+  i>  a-\-b 

q  c  -\-  d  x  +  y  a  +  b 

q  c  -{-  d  c  -\-d  x-\-y 


xi     x{yx    y{    yiZy^  2;/    2:1:^1 

X2     x%y%    y-i     y^iZi  z-i     z-ix^ 

^i    ^33/3    2/3'    2/32^3  2:3'    2:3.^3 

x^      X{y^     3/4      3/42^4  2^4      2^4:^4 

^5         ^52/5       2/5         y^^h  2^5         Z^X^ 

^<'    ^62/6    yi    2/62^6  '^i    ^&x& 

-^23  -^56  ~  ^23  -^56  ^23  ^56  ^^  -^23  ^56        ^23  -^56       -^23  ^56 

■^'i^^&l^^U  -^61  ^U^&\       •^34'^61         ^34-^61       -^^^34^61 


412 


DETERMINANTS. 


•^ m,  n  ^^^  ^m'^n         '^iv^mi 
^m,  n  =  ^myn        -^nym- 

274.  If  a,  5,  c,  d,  e,f  denote  the  six  determinants  that  can 

be  formed  from  the  array 

I  a,  13,  y,  8, 
then  will  ad  -}-  he  -}-  cf  =  0, 

275.  Prove  that 

I  ^1^2^/4 t^  I  I  ^2^3/4 4 1 

—  I  aid.,e.J^ t^  I  I  b^c^ej^ t^  | 

=  1  aAesf^ t^  I  I  Cid^eJ^. in  |. 

276.  If  An,  Bn,  Cn  are  the  inverse  elements  of  |  a-J)^c^  \  with 

respect  to  a„,  6„,  c„,  show  that 


1      1 

Ai      A2 


as 
A, 


+ 


1    1    1 

+ 

bi     62     63 

£,  -B2  ^3 

111 

Ci       C2       <?3 

Cj    62    C3 


=  0. 


277.    If  I  Ai  B2  Q{  I  he  the  reciprocal  of  |  ^i  ^2  ^3 1,  and 

«!  a2  as  ai  bi  Ci  ^  u  w^  Vi 
bi  b2  bs  ^2  ^2  C2  Wi  V  Ui 
Ci    C2    <?3       «3    bs  Cs         Vi    Ui   w 

element  for  element,  and  U,  V,  W  be  the  principal 
diagonal  elements  of  |  Ai  B2  Q  P,  then  will 
uU+vV+wW 

—  3 1  ai^2^3 1^+  ^uui+  2  vvi+  2  ww^—  6  ii^v^Wx. 


278.    If 


a 

b 

c 

d 

h 

a 

d 

c 

c 

d 

a 

b 

d 

c 

b 

a 

=  aA  +  bB  +  cC+dD, 


prove  that 


DETERMINANTS. 


413 


^  A  B  C  D  . 

B  A  D  C 

C  D  A  B 

D  C  B  A 


279.    If 


X 

y 

z 

u 

X 

y 

z 

u 

X 

ly 

u 

X 

y 

z 

u 

X 

y 

z 

h  = 

y  z  u 

X  y  z 

u  X  y 

d= 

u  X  y 

Z     11  X 

y  z   u 

prove  that 


in  which 


a  h  c 

3  _^ 

a  )8  y 

d  a  h 

8   a    ^ 

c   d  a 

y  8    a 

a 

h 

c 

d 

a 

h 

c 

d 

a 

with  similar  expressions  for  /?,  y,  8. 

280.    Given  x^  —  yz  —  a,     y'^  —  zx  =  h,     z^  - 
show  that 

(ax  +  by  +  czj-  = 


■xy=-c. 


a 

h 

c 

c 

a 

b 

b 

c 

a 

281.    If 


yz  —  u^  =  a'^^ 


vw  —  xu  =  d"^, 
prove  that 

X    w 

IV  y 

V  u 

and  solve  the  equations. 


zx  —v^=^  b"^,     xy  —  vj^^  c^, 
wu  —  yv==  e^,     uv  —  zw  =f\ 


X    W   V 

= 

10  y   u 

V    u    z 

f   b'  d' 
e'    d}    c' 


414 


DETERMINANTS. 


282. 


283. 


If  [AqB^C^ Kn  I  be  the  reciprocal  o1\a^bc^ ^"  |, 

prove  that  {—\y~''Ar/An  -=  sum  of  all  the  products 
of  b,  c,  d,  k,  taken  ti  — r  at  a  time. 


L 

M 

N 

-i- 

1 

1 

1 

I 

m 

n 

I 

m 

n 

P 

m' 

n^ 

V 

m"- 

ri 

in  which 

M=la,b,c,d)\m,l)\ 
N  =  {a,b,c,d){n,  l)\ 

284.    Prove  that 


a;   0  0 

X,' 

X^Xi 

X,- 

0  y  0 

?A^ 

y^y^ 

y? 

0   0  z 

2l^ 

Z-fl^ 

2/ 

0  2    y 

2yiZi 

yyZi  +  y.,z^ 

'ly,z. 

z    Q  X 

2z,x, 

ZiX^  +  Z.,Xi 

2z.,x^ 

285. 


y   X  0  2x,yy^  x^y^+x^y^ 

Prove  that 

X  0  0      Xi  X2 

0  y  0      y^  yi 


'Zx^y, 


=  \xy,Z2\\ 


I 
m 

n 

2p 
2q 

2r 


0  0  0  z{  z^' 

0  z  y  2y^z^  2y.,Z2 

z  0  X  2  ZiXi  2  Z2X2 

y  X  0  2x{y^  2:^,3/2 

=  -  2  A  { ^XiX,  +  m Fi Y2  +  nZ^Z^ 

+p(YA+YA) 

+  q{Z,X,  +  Z2X,) 

+  r{X,Y2  +  X2Y,)], 


in  which 
A  = 


x 

^1 

x^ 

y 

Vi 

Vi 

z 

Zi 

Z2 

Xi  is  the  minor  oix^  in  A,  etc. 


DETERMINANTS. 


415 


286.    Evaluate 


A     H    G  N  X 

HE     F  M  y 

G     F     C  L  z 

N    M    L  D  10 

X      y      z  IV  0 

in  which  A,  B,  C,  D,  etc.,  are  the  complements  of 
a,  Z),  c,  d,  etc.,  in 

a     h 
h     h 


9  ^ 

/  ^>^ 

9     f     c  I 

n     rti     I  d 


287.    If  I  ai  K,  Cs  I  =  0,  then  will 


288.    If 


X  ai  hi 

X    Ci    Gi 

— 

X  bi  Ci 

y  a.,  \ 

y    C%    CL^ 

y  ^2  ci 

z  az  h 

z   Ci  a-i 

z   h.,  Cs 

he  a  square. 


a  b  c  d  0  0 

cti  bi  Ci  di  0  0 

a^  Z^2  Ci  d^  0  0 

Q  a  b  Q  c  d 

0  ai  bi  0  Ci  di 

0  a^  bi  0  C2  di 


=  0,  then  will 


a    c    d 

— 

a    b    d 

b    c    d 

ai  Ci  di 

ai  bi  di 

bi  Ci  c/i 

a^  Ci  di 

cii  bi  di 

bi  Ci  di 

=  0. 


289.    Show  that 


0    b   c    d 

+ 

a     c    d  e 

2b   d  e  f 

Sc  e  f  ff 

+ 


a  b  c  0 
b  c  d  d 
c  d  e  2e 
d  ef  Zf 


a  0  c   d 

b  b  d  e 

c  2c  e  f 

d  Sd  f  y 

=  0. 


+  a  b  0  d 
b  c  c  e 
c  d  2df 
d  e   Se  y 


416 


DETERMINANTS. 


290.    Given  A^  =  \  knhp^Y  "^  W  ^2^3!  1 4^*1  |  k'^n^i 


292. 


A^,  =  \km^p^\ 


A,  =  I  lim2ps 


Wlim^l  14^4  1  \hnii  I 


As  =  I  hn^p^''  -^  \\  km.,  I  1 4^4 1  I  l,vi,  1 1 , 


WliTu^l  |4^3|  14^1  II,. 


find  the  value  of  AJi  +  A^k  +  A^l^  +  ^44. 

291.  Given  Un  =aiX  +  hiy  +  Ci2;,  1^22  =  h^  +  <^2y  +  t/i^^, 
Un  =biX  +  ^22/  +  e^z,  1^23  =  ^2^  +  c?iy  +  d^z, 
Wi3  =  Ci.r  +  e^y  +  C22;,       ^^33  =  ^2^  +  <^23/  +  <^32:, 

U21^^  W12,        1^31  =  Wi3,        I632  =  U23f 

Ui  =  XUn  +  3/W12  +  2:2^13, 
U2  =  XU21  +  3/W22  +  2:2^23, 

Us  =  a;w3i  +  2/W32  +  2:^33, 
C/'=  xui  +  ywa  +  2:^3, 
then  will 

I  Un,  ^22,  '^^33 1  2;^  =  4     t^n  2/12  t^i    +QU\  Un  U22  \ . 
U21  U22  U2 

Generalize. 


Un   Uu 

U, 

U21  U22 

U2 

Ui     U2 

0 

«!  a2  +  Clo  <^3  <^4  <^5  <^6  <^7  0 

«2  «3  +  <^1  <^4  +  <^0  CLb  «6  <3',7  0  0 

«3  «4  +  «2  «5  +  <^1  t)^6  +  <^0  <^7  0  0  0 

<^4  «5  +  «3  «6  +  <^2  «7  +  <3^i  «0  0  0  0 

as  ag  +  <^4  <^7  +  «3  <^2  <^1  «0  0  0 

ag    «7  +  <^5  «4  c?3     a2     ai     ^o     0 

ai  ae  as 

is  divisible  by 

<^2  —  CCo      ^3  Cti 

ag  —  aj  a4  —  ao  as 

a^  —  a2  as  ~  ai  a^  —  ao 

as  —  ag  ag  —  az  ay  —  ai 

ag  —  a4  ai  —  a^  —  at 

<^7  ~  a^  —  a4  —  as 


as  a2  ai 

a4  aa  a2  ai  c^o 


^5 

ag 
a, 

■  ai 
a2 


«6 

—  «7 

a; 

0 

0 

0 

0 

0 

ao 

0 

«i 

-ao 

Generalize  the  result  obtained. 


DETERMINANTS. 


417 


293.    Simplify 


CtiiQj22  ^12  ^^11^32  ^13^12 ^Il<^n2  <^ln<^12 


in  which  Op^  =  a^^. 


294.    If  v(a,  6,  (?)2„+i  = 


a  ^  0  0  0  0  0 

c  a  h  0  0  0  0 

0  c  a  h  0  0  0 

0  0  c  a  0  0  0 


0     0     0     0     cab 

0     0     0     0     Oca 

then  will 

V  (a,  ^,  (?)2n+i  =  «V  («^  —  2  be,  ^^  c^)„. 
295.    Also     v(^,l,l)2n  =  vG^^-2,l,l)n+v(^'-2,l,l)„_i. 


296.    lff{vi,n,p) 


prove  that 


1111 

^m    pm     ^m,     gm 

a"    ^    y'*     8^ 
a^      ^V     yP      ^P 

(a+^+y+S)/(^/^  n,^)  -  a^yS/(m-l,  n-1,^-1) 
=f{m+l,  n,p)+f(m,  n+l,p)+f(m,  7i,p+l). 


297.    If  A  =  i  a„  b,,  c„ K\  and  A  ('^'  ^' 


denote  the 


I,  7n, ) 

result  of  replacing  the  Ith,  mth,  columns  of  A 

by  a^,  fii, ,  a^,  (3m,  ,  respectively,  show  that 


Ly     771) 
cy     7vy 


X  A 


(^. /^.   

I,  m, )' 


the  determinant  on  the  left  of  "  =  "  being  of  the 
order  r. 


418 


298. 


DETERMINANTS. 

1 

1 

1 

1 

0 

0 

0 

-_ 

0 

1 

1 

1 

0 

0 

0 

0 

1 

1 

1 

0 

0 

0 

0 

1 

1 

1 

0 

0 

0 

1 

1 

1 

0 

0 

0 

1 

1 

1 

1 

0 

0 

0 

1 

1 

1 

1 

1 

0 

0 

0 

Show  that  the  value  of  a  determinant  formed  like  the 
above,  but  with  m  units  and  n  zeros  in  each  line,  is 
771  if  m.  be  prime  to  n,  but  is  zero  if  m  be  not  prime 
to  n. 

299.  If  (a,  h,  cj,  g,  h)(Ocl>,  O  +  cfy,  If  =  0, 

and  (a,  b,  cJ,  ^,  A)(c^x.  <^+X»  1)'  =  0, 

and  show  that  if 

a  =  a,  /3=b,  y=:c,  K=f,  \=g,  ii=h, 
then  will 

ac  +  h'  +  2hg~^fh=:0. 

300.  The   minors  of  order  2n—l   of  a  skew  symmetric 

determinant  of  order  2n  are  divisible  by  the  square 
root  of  the  determinant. 


V 


